DEFINITION XI- XII +

PARABOLA 

1. CONIC SECTIONS:

A conic section, or conic is the locus of a point which moves in a plane so that its distance from a fixed point is in a constant ratio to its perpendicular distance from a fixed straight line.

a) The fixed point is called the focus.

b) The fixed straight line is called the directrix .

c) The constant ratio is called the eccentrity denoted by e.

d) The line passing through the focus and perpendicular to the directrix is called the .

e) A point of intersection of a conic with its axis is called a vertex .

2. GENERAL EQUATION OF A CONIC: FOCAL DIRECTRIX PROPERTY:

The general equation of a conic with focus (p,q) and directs lx + my+ n= 0 is:

(l²+ m²)[(x - p)²+ (y - q)²]= e²(lx + my + n)²≡ ax²+ 2hxy + by²+ 2gx + 2fy + c=0

3. DISTINGUISHING BETWEEN THE CONIC:

The nature of the conic section depends upon the position of the focus S w.r.t. the directrix and also upon the value of the eccentricity e. Two different cases arise.

Case (i) : when the focus lies on the directrix:

 In this case D ≡ ABC + 2 fgh - af²- bg²- ch²= 0  and the general equation of a conic represents a pair of straight lines and if:

e> 1  the lines will be real and distinct intersecting at S.

e= 1 the lines will coincident.

e< 1 the lines will be imaginary.

Case (ii):  when the focus does not lie on the directrix :

The chronic represents :

A parabola : e= 1; D≠ 0; h½= ab

An ellipse :   0< e < 1; D≠ 0; h² < ab

A hyperbola: D≠ 0; e> 1; h² > ab

A rectangular hyperbola : e> 1; D ≠ 0 ; h² > ab ; a+ b = 0

PARABOLA 

A parabola is the locus of a point which moves in a plane, such that its distance from a fixed (focus) is equal to its perpendicular distance from a fixed line (directrix).

Standard equation of the parabola is y²= 4ax. For this parabola :

i) vertex is (0,0)

ii) Focus is (a,0)

iii) Axis is y= 0

iv) Directrix is x + a = 0

a) Focal distance:

The distance of a point in the parabola from the focus is called the focal distance of the point.

b) Focal chord:

 A chord of the parabola, which passes through the focus is called a focal chord.

c) Double ordinate:

 A chord of the parabola perpendicular to the axis of the symmetry is called double ordinate.

d) Latus rectum:

A double ordinate passing through the focus or a focal chord perpendicular to axis of parabola is called the latus rectum . For y²= 4ax.

• Length of the latus rectum= 4a.

• Length of the semi latus rectum= 2a.

• Ends of the latus rectum are L(a, 2a) and L'(a, -2a)

Note that:

i) Perpendicular distance from focus on directrix = half the latus rectum.

ii) Vertex is middle point of the focus and the point of intersection of the directrix and axis.

iii) Two parabolas are said to be equal if they have the same latus rectum.

5. PARAMETRIC REPRESENTATION:

The simple and the best form of representing the coordinates of a point on the parabola is (at², 2at). The equation x= at² and y= 2at together represents the parabola y²= 4ax,  t being the parameter.

6.  TYPE OF PARABOLA :

Four standard forms of the parabola are y²= 4ax; y²= - 4ax; x²= 4ay; x²= - 4ay

MATRICES 

DEFINITION:

A Set of mn numbers (real or imaginary) arranged in the form of a rectangular array of m rows and n columns is call ped Matrix.

ORDER OF A MATRIX 

A matrix having m rows and n columns is called a of order mxn.

a₁₁   a₁₂  a₁₃ ....a₁ⱼ ....a₁ₙ

a₂₁   a₂₂  a₂₃.....a₂ⱼ.....a₂ₙ

 |       |      |         |         |

aᵢ₁    aᵢ₂  aᵢ₃......aᵢⱼ......aᵢₙ

aₘ₁  aₘ₂ aₘ₃ ....aₘⱼ.....aₘₙ]ₘₓₙ

Or A= [aᵢⱼ]ₘₓₙ, 1≤ i≤ m, 1 ≤ j≤ n, i,j ∈ N

TYPES OF MATRIX 

1. Rectangular matrix : If the number of rows in the columns of a matrix are not equal, then the matrix is called rectangular Matrix, i.e., m≠ n

2. Row Matrix : A matrix containing only one row is called a row matrix. i.e., m= 1

3. Column Matrix: A matrix containing only one column is called a column matrix i.e., n= 1

4. Null or Zero Matrix : If every element of a matrix is zero, then it is called a zero or null matrix.

5. Square matrix: A matrix in which the number of rows is equal to the number of columns is called square Matrix p.

6. Diagonal Matrix: A square Matrix in which every non diagonal element is zero is called a diagonal Matrix.

7. Scalar matrix : A square Matrix in which all the elements in the diagonals are equal and rest of the elements are zero is called a scalar matrix.

8. Identity or unit matrix: A square in which each diagonal elements is unity and all other elements are zeros is called a unit matrix. It is denoted by I.

9. Upper triangular matrix : A square Matrix in which all the elements below the diagonal are zeros is called an upper triangular matrix.

10. Lower triangular matrix: A square Matrix in which all the elements above the diagonal are zeros is called a lower triangular matrix.

11. Equal matrix : Two matrices are said to be equal if they are of the same order and each element of one matrix is equal to the corresponding element of other matrix.

OPERATION ON MATRICES 

• Matrix addition : Let A= [aᵢⱼ] and B=[bᵢⱼ] be two mxn matrices. Then the sum A+ B of A and B is defined as A+ B= [cᵢⱼ]ₘₓₙ , where cᵢⱼ = aᵢⱼ + bᵢⱼ for each i and j.

• Substraction of a matrix : Let A= [aᵢⱼ]ₘₓₙ and B=[bᵢⱼ]ₘₓₙ be two matrices of same order, then the difference A - B of A and B is defined as A - B = A +(- B)= [aᵢⱼ - bᵢⱼ]ₘₓₙ for each i and j.

• Negative of Matrix : Let A= [aᵢⱼ]ₘₓₙ be a matrix, then the negative of the matrix A is defined as the matrix [-aᵢⱼ]ₘₓₙ and is denoted by - A. Thus negative of a matrix is obtained by making all the elements of the given matrix negative .

• Scalar multiplication of a matrix: Let [aᵢⱼ]ₘₓₙ be a matrix and k is a scalar, then the Matrix obtained by multiplying the individual element of the matrix A by k is called the scalar multiplication of A by k and is denoted by kA or Ak.

• Multiplication of two matrices : Let A= [aᵢⱼ]ₘₓₙ and B=[bᵢⱼ]ₙₓₚ be two matrices, then the product AB= C = [cᵢₖ]ₘₓₚ where cᵢₖ = aᵢ₁ b₁ₖ + aᵢ₂ b₂ₖ + ....+ aᵢₙ bₙₖ.

Properties of matrix Addition and Multiplication 

• Commutative law

If A and B are two matrices, then 

A+ B= B+ A and AB ≠ BA 

•Associative law

If A, B and C are three matrices, then 

(A+ B)+ C = A+(B + C) and (AB)C = A(BC(

• Distributive law

A(B+ C)= AB+ AC and (A+ B)C = AC+ BC 

• Existence of Identity

 The null matrix is the identity element for Matrix edition i.e., A+ O = A= O+ A

If A is a square Matrix, then IA = A = AI, where I is identity matrix of same order.

• Existence of inverse :

For every matrix A= [aᵢⱼ]ₘₓₙ there exists a matrix [-aᵢⱼ] ₘₓₙ denoted by - A, such that A+ (-A)= O= (-A)+ A

• If A is mxn matrix and O is a null Matrix, then

i) Aₘₓₙ Oₙₓₚ = Oₘₓₚ

ii) Oₚₓₘ Aₘₓₙ = Oₚₓₙ 

TRANSPOSE OF A MATRIX 

• Let A= [aᵢⱼ]ₘₓₙ matrix. Then the transpose of A, denoted by Aᵀ or A', is an nxm matrix such that Aᵀ=  [aᵢⱼ]ₘₓₙ and i= 1,2,....m and j= 1,2,....,n.

Thus Aᵀ is obtained from A by changing its rows in to columns and its columns into rows.

Properties of Transpose 

• Let A and B be two matrices, then 

i) (Aᵀ)ᵀ= A

ii) (A+ B)ᵀ= Aᵀ + Bᵀ,where A and B being of the same order.

iii) (kAᵀ)= kAᵀ, k  be any scalar (real or Complex)

iv) (AB)ᵀ = BᵀAᵀ , A and B being conformable for product AB . (Reversal law)

v) (ABC)ᵀ= CᵀBᵀAᵀ.

ELEMENTARY OPERATION 

There are six operations (transformations) on a matrix , three of which are due to rows and three due to columns, which are known as alimentary operations or transformations .

i) The multiplication of the elements of any row or column by a non zero number. Rᵢ --> Rᵢ

The corresponding column operation is denoted by Cᵢ --> kCᵢ where k≠ 0

For example, applying R₂ --> (1/4) R₂ to A

A= a₁₁   a₁₂  we get a₁₁      a₁₂

      a₂₁   a₂₂              a₂₁/4  a₂₂/4

ii) The interchange of any two rows or two columns i.e., Rᵢ <---> Rⱼ and Cᵢ <--> Cⱼ.

For example, applying R₁ <--> R₃ to A

     a₁₁   a₂₁   a₁₃               a₃₁   a₃₂   a₃₃

A=a₂₁   a₂₂    a₂₃ we get  a₂₁   a₂₂   a₂₃

     a₃₁   a₃₂    a₃₃               a₁₁   a₁₂   a₁₃

iii) The addition to the elements of any two rows or column, the corresponding elements of any other row or column multiplied by any non-zero number 

i.e., Rᵢ---> Rᵢ + kRⱼ or Cᵢ---> Cᵢ + kCⱼ.

For example applying C₁--> R₁ - kR₂ to A

A= a₁₁   a₁₂  we get a₁₁ - ka₂₁  a₁₂ - ka₂₂

      a₂₁   a₂₂              a₂₁            a₂₂

 

SOME SPECIAL MATRICES 

• symmetric matrix : A square Matrix A= [aᵢⱼ] is called a symmetric matrix if  aᵢⱼ = aⱼᵢ for all i, j.

• Skew-symmetric matrix: A square matrix A= [aᵢⱼ] is a Skew-symmetric matrix if  aᵢⱼ = - aⱼᵢ for all i, j.

• Orthogonal matrix : A square matrix A is called an orthogonal matrix if AAᵀ =AᵀA = I.

• Equivalent matrices: Two matrices A and B are equivalent if one can be obtained from the other by a sequence of elementary row transformations .

INVERTIBLE MATRICES 

if A is a square matrix of order n and if there exists another square Matrix B of the same order n, such that AB = BA= I, then B is called the inverse matrix of A and it is denoted by A⁻¹.  In that case A is said to be invertible.

Uniqueness of inverse 

Inverse of a square matrix, if it exists , is unique .

Let A=[aᵢⱼ] be a square matrix of order n. If possible, let B and C be two inverses of A.

To show: B= C

Since B is the inverse of A

=> AB= BA = I

Since C is also the inverse of A

=> AC= CA= I

Thus B= BI = B(AC)= (BA)C = IC = C

If A and B are invertible matrices of the same order, then (AB)⁻¹= B⁻¹A⁻¹.

Inverse of a Matrix by Elementary Operation 

To find A⁻¹.using elementary row operations , write A= IA and apply a sequence of row operations on A= IA till we get, I= BA. The matrix B will be the inverse of A. Similarly, if we wish to to find A⁻¹ using column operations , then write A= AI and apply a sequence of colum operations on A= AI till we get, I= AB 

DETERMINANTS 

DEFINITION: Corresponding to each square Matrix A= (aᵢⱼ)ₘₓₙ there is an associated expression , called the determinant of A, denoted by det A or |A|, written as 

|a₁₁   a₁₂  a₁₃ ....a₁ₙ

a₂₁   a₂₂  a₂₃.....a₂ₙ

....    ....   ... ...   ...

aₙ₁  aₙ₂   aₙ₃ ....aₙₙ|ₘₓₙ

• Value of determinant of order 1

Let A= [a] be Matrix of order 1, then determinant A is defined to be equal to a.

•  Value of determinant of order 2

|a₁₁   a₁₂ 

 a₂₁    a₂₂| = (a₁₁ a₂₂ - a₂₁ a₁₂)

 

• Value of a determinant of order 3 or more 

The value of a determinant is the sum of the product of a row (or a column) with their corresponding factors.

So, det A= ⁿᵢ₌₁∑ aᵢⱼ Cᵢⱼ 

 

• Properties of determinant

i) det Iₙ= 1, where Iₙ is unit/identity where unit/identity matrix of order n.

ii) det 0ₙ= 0, 0 is null Matrix (square matrix of any order).

iii) A= (aᵢⱼ)ₙₓₙ  then |A|= |A'| (Reflection Property).

iv) det (AB )= det A . det B, where A & B are matrices of same order.

v) det(kA)= kⁿ det A, if A is of order n x n.

vi) det (Aⁿ)= (det A)any if n ∈ I⁺

vii) |A|= 0 iff

      a) Any two rows or columns are identical.

      b) Any two rows or columns are in proportion.

      c) Each elements of any row/ column is zero.

viii) if each element of a row/column of a determinant is multiplied by k then value of new determinant is k times the original Determinants.

ix) Determinant of a diagonal matrix is the product of its diagonal elements.

x) If two row/columns of a determinant are interchanged, then the determinant retains its absolute value but its sign is changed.

xi) if each element of a row/columns of a determinant is expressed as a sum of two or more terms, then determinant can be expressed as the sum of two or more determinants.

xii) if any row/column of a determinant, a multiple of another row/column is added, then the value of determinant does not change.

xiii) The sum of product of the elements of any row/column of a determinant with cofactors of the corresponding elements of any other row/ column is zero.

APPLICATION OF DETERMINANTS 

• Area the triangle with vertices A(x₁, y₁), B(x₂, y₂) and C(x₃, y₃) is given by 

(1/2) | x₁      y₁     1

            x₂      y₂     1

            x₃      y₃      1

• If area is zero then there points are collinear.

MINORS AND COFACTORS 

• Minor of  aᵢⱼ in |A|: The minor of an element aᵢⱼ in |A| is defined as the value of the determinant obtained by deleting the iᵗʰ row and jᵗʰ column of |A|, and it is denoted by Mᵢⱼ .

• Cofactor of aᵢⱼ in |A|: The cofactor Cᵢⱼ of an element aᵢⱼ is defined as Cᵢⱼ = (-1)ⁱ⁺ʲ Mᵢⱼ.

ADJOINT OF A MATRIX 

• Adjoint of a matrix A= (aᵢⱼ)ₙₓₙ is defined as adj A= [Cᵢⱼ]ₙₓₙ, where 

Cᵢⱼ represents cofactor of aᵢⱼ in |A|.

Properties of Adjoint of a matrix

• For every square matrix

A(adj A)= (adj A)= |A|. I.

• If |A|=0 matrix A is called singular else non singular.

•  if A and B are non singular matrices of the same order, then AB and BA are also non singular matrices of same order.

• (adj AB)= (adj B).(adj A)

• (adj A)' = adj A'.

• let A be n x n matrix, then 

i) |adj A|= |A|ⁿ⁻¹

ii) adj (adj A)= |A|ⁿ⁻²A

iii) |adj (adj A|= |A|(ⁿ⁻¹)^2

INVERSE OF A MATRIX 

A non zero matrix A of order n is said to be invertible if there exists a square matrix B of order n such AB = BA= I. We say A⁻¹= B.

• A matrix A is invertible if |A|≠ 0.

• A⁻¹= 1/|A|.   (adj A).

• If AB= AC => B = C if |A|≠ 0.

• (AB)⁻¹= B⁻¹A⁻⅓.

• (Aᵀ)⁻¹= (A⁻¹)ᵀ.

• If A is an invertible symmetric matrix then A⁻¹ is also symmetric.

SOLUTION OF SYSTEM OF EQUATIONS

• let AX = B be the given system of equations:

    i) If |A|≠ 0, the system is consistent and has one unique solution.

   ii) If |A|= 0 and (adj A)B ≠ 0, then the system is inconsistent so it has no solution.

   iii) If |A|= 0 and (adj A)B=0, then the system is consistent but has infinitely many solutions. 

TANGENT & NORMAL 

Things to remember:

CIRCLE 

Circle is defined as a local self found in the Cineplex as the distance from a fixed point in the plane is constant the fixed point is called the centre of the circle on the constant distance is called the radius of the circle standard equation of a circle Central forms represent the circle with centre in the radius of the centre is at 007 circle become circle passing through the origin circle touching circle passing through the origin and Central line on the exactis and circle passing to the original Central line on the y-axis general equation the circle is equation of a circle centre in the radius and for a real circle and for the Point Circle and forum in imaginary circle conditions of a circle it is a second degree Celsius equation there is no time containing it contest we are literally constant for the circle refers to write as Centre is under radius equation of a circle when coordinates are pain the point of the time which are position of a point a circle the point lies outside on or inside a circle according circleall the pragmatic position of we write the points are the parameter equation normal atomic count On the circle is a straight line which is perpendicular to the tangent with the curve at the point length of the tangent from to the circle pair of tangent drawn from a point to a given circle is given by circle is similarly for the circle director circle the locus of the point of intersection at 2 perpendicular timing to a given circle is called a director circle is a director circle for circle the condition that to circle should interface area and sufficient condition for the surprise to interested to distinct points is where with the centres with the radii of the two circles external and internal contact of the circles to circle with centre and ready respectively touch each other internallength of an external or direct common time zone or transversal common timing to length of a external on direct common tangent and internal transfer common tangent to the two circle is given by were distance between the centre of the circles are there of the two circles the length of the internal common tangent is always less than length of the external or direct common tangent the direct common timing to circle the transversion common Tangled also meet on the line centre and divided internal on the ratio of the riding chord of the contact of two tangents drawn from to the circle similarly for the circle equation the chord with given midpoint is for the circle equation of the chords with midpoint similarly for the general circle common code the to circle the core joining the points of intersection of two given circle is called their common cause if denote the centre of the given circle then the order radius of the circle length of the perpendicular from the common cause of the circle to circle are safety after 1 time to be circles at either point of section are at right anglecondition for two circles the orthogonal is ID collecting the focus in such a way so that length of tangent drawn from it to the two circles are equal is straight line known as the radical accessed of the given to circles a part 2 circles the radical exercise given by any circle passing through the point of intersection of circle where is constant any circle passing through a point of intersection

PROBABILITY 

Random experiment and operation which can result in any one app its weight outcomes and outcom cannot be predicted its certainty sample space on a sample point the set of all possible outcomes around experimental sample space and is denoted by a brief possible outcomes April point trial and experimental repeat under the similar code does not give the same result it's time a subset sample space setup some possible outcomes a random experiment simple in bank contains only one sample points in banks contains no sample points missed compound more than one sample point complement of an agent contents all the sample point other than those content in event the union and the consist of all points are both the intersection and concerts of all points that are in both and and have a no points in common we write the complement for consist of all the points of not in the sequel like less none of them is expected to occur in preference in othersmutually exclusive events occurrence of one of them precludes are current affairs exhaustiblements performance of the experiment result in occurrence of at least one of them binomial distribution out of independence trials the probability of getting success at least success at most success where is the probability of success and probability classical definition basicap probability for an event is complement in sample space for event for exhaustive banks in a sample space for mutually exclusive a balance in a sample space for mutual exclusive and exhaustive events for an event in sample space the odds in favour where is the complement of the event also against conditional probability the conditional probability of given is independent event events are independent if the reference does not affect each other and also they must be there wise independent total probability in the best theorem in a sample space to be the set of mutual explosion exhaustive set of sample space the total probability of a band is given by the an event of which has already occurred than condition probability of occurrence of any one of the event out of 1 to

RELATION AND FUNCTION

EXERCISE - A

1) Write down the relation shown by the diagram,

by listing the ordered pairs. State the domain, co-domain, and the range of the relation .

2) Which of the following are relation from B to A, where A={a,b,c,d} and B={x, y,z} ?

i) {(z,x),(z,y),(x,a)} 

ii) {(z,a),(z,b),(z,c),(z,d)}

iii) {(x,a),(y,b)}

iv) {(b,y),(c,z),(a,x)}

v) {x,d),(y,c),(z,b)}

3)  In each of the following state which of the ordered pairs belong to the given relation ?

i) {(x,y): x > y+ 5} ; (1,0),(8,2),(0,1),(2,8),(9,3), (10,7), (123,4)

ii) {(x,y): xy= 12} ;  (3,4),(4,3),(12,0),(0,12),(12,1),(6, 2),(7,5).

iii) {(x,y: y= (x +3)/(x -1), x ∉ 1}; (0,1),(2,5),(5,2),(3,3),(7,5),(7,5/3).

4) Let N be the set of natural numbers. Describe the following relations in words, giving their domain and the range.

a) {(2,1),(4,2),(10,5),(18,9),(20,10)}

b) {(3,1),(6,2),(15,5)}

c) {(1,4),(5,16),(7,22),(12,37)}.

5) Z is set of integers. Describe the following relation in set builder form, given its domain and range.

{(0,-7),(2,-5),(4,-3),(-13,-20),....}

6) Write down the domain and range of the relation (x,y): x =3y and x and y are natural numbers less than 10.

7) Determine the domain and range of the relation R.

a) R={(x +1),(x +5): x ∈ {0,1,2,3,4,5}

b) R = {(x, x³) : x is a prime number less 10}

8) Given A={-2,-1,0,1,2}, list the ordered pairs determined by each of the following relations applied on A:

i) R₁= 'is less than '

ii) R₂ = " is the square of".

iii) R₃ = "is the additive inverse of".

iv) R₄ = 'is equal to '.

9) Given A= {2,3,4,5,6}. List the element of each of the following relations:

i) {(x,y) ∈ A x A : x = y}.

ii) {(x,y) ∈ A x A : x > y, x/y ∉ W}.

iii) {(x,y) ∈ A x A : x is a divisor of y and x ≠ y}

EXERCISE - B

1) Consider the following properties of relations : Symmetric(S), Transitive(T), Reflexive (R), Equivalence (E), None of these (N).

State which property/properties are satisfied by each of the following relations :

(Give the answers in terms of S, T, R, E and N).

a) "is greater than" for the set of real numbers.

b) "is the cube of" for the set of all real numbers.

c) " is the sister of" for a set of children.

d) " is similar to" for the set of triangles.

e) " is parapendicular to"  for a set of co-planar lines.

2) Write down a relation which is 

a) Only transitive 

b) Only symmetric .

c) Only reflexive and transitive.

d) Only symmetric and reflexive.

3) Prove that if A is the set of the members of a family and R means " is brother of" then it is a transitive relation.

4) Show that the relation R in the set {1, 2, 3} given by R={(1,2),(2,1)} is symmetric but neither reflexive nor transitive.

5) Let R be the relation in the set =1,2,3,4} given by {(1,2),(2,2),(1,1),(4,4),(1,3),(3,3),(3,2)}. Choose the correct answer:

a) R is reflexive and symmetric but not transitive.

b) R is reflexive and transitive but not simmetric.

c) R is symmetric and transitive but not reflective.

d) R is an equivalence relation.

6) Show that the relation R in the set A of real numbers defined as R= {(a,b): a ≤ b} is Reflexive and transitive but not symmetric.

7) Show that the relation R in the set A of real numberr defined as R ={(a,b): a ≤ b²} is neither reflexible nor symmetric nor transitive .

8) In the set of all triangles in a plane, show the relation of similarity is an equivalence relation.

9) Is the relation "is the square of" for the set of natural numbers N an equivalence relation ?

10) Consider the following properties of relations : symmetric (S), Transitive (T), Reflexive (R) and Equivalence (E).

A relation may may have all, some or none of these properties. For each part, state whether the relation has some or all the properties mentioned by writing the letters S, T, R, E in the space provided. Write N, if none of the properties satisfy .

a) "is smaller than"

b) " is the father of".

c) " is parallel to" for set of straight lines.

d) " is a multiple of" for a set of positive integers .

e) "is congruent to".

11) Answer true or false :

The relation "is congruent to" in a set of triangles is a transitive relation.

12) if R is a relation in N x N defined by (a,b) R (c,d) if and only of a+ d = b + c, show that R is an Equivalence relation.

13) Prove that the relation R in the set A={1,2,3,4,5} given by R={(a,b): |a - b| is even} is an Equivalence relation.

14) Let l be the set of all integers and R be the relation on I defined by aRb iff (a + b) is an even integer for all a, b ∈ I. Prove that R is an equivalence relation.

15) Let I be the set of all integers and R be the relation on I defined by R = {(x,y): x, y ∈ I, x - y is divisible by 11}, Prove that R is an Equivalence relation.

FUNCTION 

EXERCISE - A

1) Which of the following relations are functions ? Give reasons , In case of a function, determine its domain and range.

i) {(1,-2),(3,7),(4,-6),(8,11)}

ii) {(1,0),(1,-1),(2,3),(4,10)}

iii) {(a,b),(b,c),(c,d),(d,c)}

2) State whether or not each of the following diagrams defines a function of 

 A={a,b,c} into B={x,y,z}.

3) Which of the following relations are functions?

i) 3x +2.

ii) a is the capital of b ∈ B and B is the set of all countries, a∈ A  and A is the set of capital of cities of countries.

iii) y < x +3.

iv) y is the Maths teacher of x where x represents any pupil taking up Maths in a school.

v) y is a Maths pupil of x, where x represents any Maths teacher in a school.

4) State the domain of these functions:

a) f: x ---> 5x

b) g: x ---> 5x, x ∈ Z

c) h: x ---> 2/(x -7).

d) F: x ---> 5x, x ∈ {0,1,2}

e) f: x ---> x/5

f) F: x ---> 6/x.

g) H: x ---> x²+ 5x - 6.

h) g: x ---> (x- 4)/{(x -3)(x +6)}.

i) g: x ---> x/1, x ∈ {2,4,6}

j) g: x ---> 1/x, x ∈ R.

EXERCISE - B

1) A function f given as f: {(2,7,(3,4),(7,9),(-1,6),(0,2),(5,3).

Is this function one-one onto ?

Interchange the order of the elements in the ordered pairs and form the new relation. Is this relation a function ? If it is a function, is it one-one onto.

2) Determine if each function is one-one .

i) To each person on the Earth assign the number which corresponds to his age.

ii) To each country in the world assign the latitude and longitude of its capital.

iii) To each book written by only one author assign the author.

iv) To each country in the world which has a prime minister assign its prime minister.

3) Let f: A--> B. Find f(A), i.e., the range of f, if f is an onto function.

4) Show that the function f: R---> R given by f(x)= cosx for x ∈ R, is neither one-one nor onto.

5) Let A={-1,1}. Let functions f, g and h of A onto A be defined by:

i) f(x)= x.

ii) g(x)= x³

iii) h(x)= sinx.

Which function, if any, is onto?

6) Given A={2,3,4}, B={2,5,6,7}, construct an example of each of the following.

i) A one-one mapping from A to B.

ii) A mapping from A to B which is not one-one .

iii) A mapping from B to A .

7) Are the following ssta of ordered pairs functions ? If so, examine whether the mapping is onto or one-one.

i) {(x,y): x is a person, y is the mother of x}

ii) {(x,y): a is a person, b is an ancestor of a}

8) Is the function f: N --> N (N is the set of natural numbers ) defined by 

f(n)= 2n +3 for all n ∈N onto ?

9) let A={x =0 ≤ x ≤2} and B={1}. Give an example of a function from A to B. Can you define a function from B to A which is onto ? Give reasons for your answer.

10) Prove that the function f: R ---> R, f(x)= x²+ x is a many-one into function.

11) Let A={1,2,3}, B={4,5,6,7} and let f={(1,4),(2,5),(3,6)} be a function from A to B.

Show that f is one-one but not onto.

12) Show that the function f: R---> R : f(x)= 3- 4x is one-one onto and hence bijective.

EXERCISE - C

1) If the function f: N --> N is defined by f(x)= √x, then find f(25)/{f(16)+ f(1)}.

2) If f(x)= x³/2 - x²/2 + x - 16, find f(1/2).

3) If f(x)= 7x⁴- 2x³- 8x -5, find f(-1).

4) If f(x)= 3x -2 when x ≤ 0

                  x +1 when x > 0

Find f(-1) and f(0).

5) If f(x)= log{(1- x)/(1+ x)}, show that f(a)+ f(b)= f{(a+ b)/(1+ ab)}.

6) If f(x)= 2x √(1- x²), then show that f{sin(x/2)} = sinx.

7) If f(x)= cos(log x), then show that 

f(1/x). f(1/y) - (1/2)[f(x/y) + f(xy)]= 0.

8) If y= f(x)= (5x +3)/(4x -5), then show that f(y)= x.

9) State with reason whether the following functions have inverse.

i) f= {1,2,3,4} --> {10} with f= {(1,10),(2,10),(3,10),(4,10)}

ii) g={5,6,7,8} --> {1,2,3,4} with g={(5,4),(6,3),(7,4),(8,2)}

iii) h={2,3,4,5}--> {7,9,11,13} with h={(2,7),(3,9),(4,11),(5,13)}

10) Let A={3,5,7,9}, B={9,25,49,81} and f: A--> B be given by f(x)= x². Write f and f⁻¹ as a set of ordered pairs.

11) Consider f: {1,2,3}--> {a,b,c} given by f(1)= a, f(2)= b and f(3)= c. Find the inverse f⁻¹. Show that (f⁻¹)⁻¹= f.

12) Consider f: R---> R given by  4x +3. Show that f is invertible and find the inverse of f.

13) Let S={a,b,c} and T={1,2,3}. Find D⁻¹ of the following function F from S to T, if it exists,

i) F={(a,3),(b,2),(c,1)}

ii) F={a,2),(b,1),(c,1)}

14) f= R --> R is a function defined by f(x)= 10x -7. If g= f⁻¹, then find g(x).

15) If f(x)= (1+ x)/(1- x), then show that f⁻¹(y)= (y- 1)/(y +1).

16) If f(x)=(3x +2)/(5x -3), then show that f⁻¹(x)= f(x).

17) If R---> R defined by f(x)= (3x +5)/2 is an invertible function. Find  f⁻¹.

18) Consider f: R⁺---> [4, ∞) given by f(x)= x²+ 4. Show that f is invertible with the inverse of f⁻¹ of f given by f⁻¹(y)= √(y - 4}, where R⁺ is the set of all non-negative real numbers.

19) If f(x)= x⅖+ Kx +1, for all x and if it is an even function, find k.

20) If f(x)= x³- (k -2)x²+ 2x, for all x and if it is an odd function, find k.

21) Is there a function f which is both even and odd ?

22) The function f(x)= log(x + √(x²+1)), is 

i) an even function 

ii) an odd function 

iii) a periodic function 

iv) Neither an even nor an odd function.

23) Show that (1/x) log[√{x + √(x²+1)}] is an even function.

EXERCISE - D

1) Find (gof)(3), (fog)(1) and (fof)(0) if

a) f(x)=3x -2, g(x)= x².

b) f(x)= |x +2|, g(x)= - x²

c) f(x)= x²-1, g(x)= √x.

2) If f(x)= x +5 and g(x)= x²-3, find the following:

a) f(g(0))

b) g(f(0))

c) f(g(x)).

d) g(f(x)).

e) f(f(x)).

f) g(g(x)).

g) f(f(-5)).

h) g(g(2)).

3) If u(x)= 4x -5, v(x)= x², and f(x)= 1/x, find 

a) u(v(f(x))).

b) u(f(v(x)))

c) f(u(v(x))).

4) Find the inclined values, where 

g(t)= t²-1 and f(x)= 1+ x.

a) g(f(0))+ f(g(0)).

b) g(f(2) +3)

5) If f(x)= (2x +1)/(3x -2), then (fof)(2) is equal to 

a) 1 b) 2 c) 3 d) 4

6) If f(x)= (1- x)/(1+ x), then f(f(cos 2θ) is equal to 

a) cos2θ b) tan2θ c) sec2θ d) cot2θ

7) If f(x)= sin²x and the composite function g(f(x))= |sinx|, then the function g(x) is equal to 

a) - √x b) √x c) √(x -1) d) √(x +1)

8) If f: R---> R and g: R---> R are defined respectively as f(x)= x²+ 3x +1 and g(x)= 2x -3, find 

a) fog 

b) gof 

9) If f: R --> R, f(x)= x², g : R---> R, g(x)= cosx. Find 

a) fog 

b) gof 

And show the fog ≠ gof 

10) If f(x)= logx, g(x)= x³, then show that f(g(a))+ f(g(b))= 3f(ab).

11) If f(x)= (25 - x⁴)¹⁾⁴ for 0< x < √5, then show f[f(1/2)]= 2⁻¹·

APPLICATION OF DERIVATIVES

MOTION IN A STRAIGHT LINE (OR RECTILINEAR MOTION)

EXERCISE - A

1) A particle is moving in straight line in such a way that its distance in a centimetres from a fixed point on the line after t seconds, is given by s= 5t³- 2t +3. Find the distance, velocity and acceleration at the end of 2 seconds.      39cm, 58 cm/s, 60 cm/s²

2) A particle moves along a line according to the formula, s= t³+ at²+ bt + c, where a,b,c are constants . Find a,b,c if at t= 1s, we have s= 3m, velocity= 7 m/s and acceleration= 12 m/s².      3,-2,1

3) The displacement s in metres of a particle moving in a straight line in t seconds is given by s= 45t + 11t²- t³. Find the time when the particles comes rest.     9sec

4) The distance s metres moved by a particle travelling in a straight line, in t seconds, is given by, s = 3t + t³. Calculate 

a) the acceleration after 2 seconds.     12m/s²

b) the time when the velocity is 30 m/s.     3sec

c) the distance moved during the 3rd second.        22m

5) A particle moves in a straight line in such a way that its position at a time t is given by x= 3t + cos3t. Show that its velocity and acceleration both vanish together. Find, how far it moves before coming to rest.      π/2

6) The distance x metres covered by a particle moving along a straight line in t seconds is given by formula x= 2t³- 9t²+ 5t +8.

Find when its acceleration is zero. Also, find the velocity at the instant.     1.5 sec, -8.5 m/s

7) A particle moving according to the formula s= 6 + 20t - t², starts from a distance of 6m from a mark and moves along a line farther and farther from the mark. How far from the mark does it go, before it starts moving the opposite direction ?    106m

8) if a particle moves in a straight line according to the formula s= t³- 6t²- 15t; find the time interval during which the velocity of negative and acceleration is positive.     t=. 2 to t= 5

9) A particle moves along a line according to the law, x= √(at²+ 2bt + c), where a,b,c are constant. Prove that the acceleration varise inversely as the cube of the distance.    

10) A particle moves in a a line so that so that x= √t. Show that the acceleration is negative and proportional to the cube of the velocity.

11) A point moves in a line so that its distance s from a fixed point at any time t is proportional to tⁿ, if v be the velocity and f the acceleration at any time t, show that v²= nfs/(n -1).

12) The law of motion of a particle moving in a line given by s= vt/2, show that the acceleration is constant.

13) A car is running on a straight road. The distance travelled and the time taken are  connected by the formula s= (t²- 2t), where s is measured in kilometres and t in hours. When the reading of the odometer is 63, what is the reading of the speedometer?    16

RAW- A

1) A particle is moving in a straight line and the distance s cm covered by it in t seconds is given by the formula, s= 20t - 4t²

 Find the distance, velocity and acceleration at the end up 2 seconds.   24cm, 4cm/s, 8 cm/s²

2) A particle, starting from a fixed point O, moves in a straight line. The distance s metres described by it in t seconds is given by, s= 11+ 5t + t².  Find 

a) the distance from O after 4 seconds.    95m

b) its speed after 4 seconds.       53m/s

c) its acceleration after 4 seconds.      24 m/s²

d) distance travelled by it in the 4-th second.      42m

3) The displacement x of a particle at a time t is given by, x = 2t³- 5t² + 4t+3.  Find 

a) the time when the acceleration is 8 cm/s².      1.5s

b) velocitya and displacement at the instant.      2.5 cm/s, x= 4.5 cm

4) A particles moves along a straight line according to the law, s= t²- 6t²+ 19t -4. Find 

a) its displacement and acceleration when its velocity is 7 m/s.     18m, a= 0

b) its displacement and velocity when its acceleration is 6 m/s².     26m, 10m/s

5) A particle moves a straight line so that after t second its distance from a fixed point O on the line is s meters, where s= t³- 4t²+ 3t. Find 

a) when the particle is at O.         0, 1, 3

b) what are its velocity and acceleration at these instants.     3m/s, -2m/s, 6m/s and -8m/s², -2m/s², 10m/s²

6) A particle moves along a line according to the law, s= at²- 2bt + c, where a,b,c different are constants . Prove that the acceleration is constant.

7) If a particle moves along a line so that the distance described is proportional to the square of the time of description, prove that the velocity is proportional to the time and the rate of increase of the velocity will be constant.

8) A car from rest and moves a distance s metres in seconds , where s= a cos t + b sin t. Show that the declaration at time t is the negative of the distance covered by it in t seconds.

9) The distance s in metres described by a particle in t seconds is given by, s= (aeᵗ + be⁻ᵗ). Show that the acceleration of the particle at time t is equal to the distance travelled by it upto time t.

10) A particle moves in a line according to the law s= at²+ bt + c, where a, b, c are constant and s is the distance of the particle from a fixed point O, covered in t seconds. Initially, the particle is 10cm away from O and its initial velocity is 12 cm/s. If the particle moves with a uniform acceleration of 4cm/s²,  find the distance travelled by it in the 7-th second.        38m

11) The displacement x metres of a particle moving in a straight line at a time t seconds is given by, x = 2t³- 9t²+ 12t +1.  find 

a) the velocity and acceleration at t= 1 second.    0, -6m/s²

b) the time when the particle stops momentarily.     1s, 2s,

c) the distance between two stops .     6m

12) A particles moves in a line according to the law, s= 6t³+ 20t²+ 9t, where s is in centimetres and t in seconds. Find the initial velocity and acceleration of the particle.      9m/s, 40 cm/s²

13) A particle is moving on a line according to the law, s= tan⁻¹t + at²+ bt + c, where a, b, c are constants. it is given that 1.5 m/s². Find the values a,b,c.     1, 1/2, (8-π)/4

MOTION UNDER GRAVITY 

EXERCISE - B

1) A Stone is thrown vertically upwards moving in a line, Its equation of motion is, s= 29.4t - 4.9t².  Find the maximum height reached by it.     44.1 units 

2) The maximum height is reached in 3 seconds by a stone thrown up vertically and moving under the equation s= ut - 4.9t², where s is in metres and t is in seconds. Find the value of u.       29.4

3) A ball thrown vertically upwards, falls back to the ground after 8 seconds. If the equation of motion is s= ut - 4.9t², where s is in metres and t is in seconds, find the velocity at t= 1.         29.4 m/s

4) Two stones are thrown up simultaneously. The equations of motion are s= 19.6t - 4.9t² and s= 9.8t - 4.9t² for the first and second stone respectively. What is the height of the second stone, when the height of the first stone is maximum ?      to the ground 

5) Someone standing on a tower of height 19.6 metres , throws a stone vertically upwards. it moves in a vertical line slightly away from the line of the tower and falls on the ground. If its equation of motion is, s= 19.6t - 4.9t², where s is in metres and t is in seconds, how much time does it take for the upward motion and how much for the downward motion ?        2, 4 sec

6) The motion of a ball thrown vertically upwards satisfyies the equation. s= bt½+ ct,  Where s and t are measured in metres and seconds respectively. if the maximum height reached by the ball is 4.9 metres and it acceleration is -9.8 m/s², find its height after half a second.      3.675m

7) A stone is thrown upwards on certain planet and its equation of motion is s= 5t - 3t², where s is in metres and t is in seconds. What is the acceleration due to gravity of the planet ? After what time will the stone fall back on the planet again ?     5/3 sec

8) A ball is freely from the top of a tower and during the last second of its motion, it is observed to fall (16/25) of the total height. If the equation of motion is, s= 4.9t², where s in metres and t is in seconds ; find the height of the tower.   30.625 m

RAW- B

1) A stone thrown vertically upwards has its equation of motion, s= 49t - 4.9t¹,  where s is in metres and t in seconds. Find the maximum height reached by it.     122.5m

2) A particle is moving in a vertical line as per equation, s= 100t - 4.9t², where s is in metres and t in seconds. Find its velocity at t= 1. At what time is its velocity zero ?

What is the maximum value of s ?     90.2 m/s, 10.2s, 510.2m

3) An arrow shot vertically upwards, movey according to the formula, where s= 49t - 4.9t², wherrs is in metres and t in seconds. Find the time taken by it to reach a height of 117.6 metres .

What is its velocity at the end of 8 second ?

After how much time will it fall on the ground ?    4s,6s, -29.4 m/s, 10s

4) A ball projected vertically upwards has its equation of motion, s= ut - 4.9t², where s is in metres and t in seconds . If the maximum height reached by the ball is 44.1 metres ,  find the value of u.      29.4

5) A short fired vertically upwards is known to be at a point A at the end of 2 seconds and also again there after 3 more seconds. if the equation of motion is s= ut - 4.9t², where s is in metres and t in seconds; find the height of A above the point of projection.      49m

6) A particle falls down freely from the top of a tower and in the last second of its motion it falls down (9/25) of the total height of the tower. If the equation of motion is s= 4.9t²,  where s is in metres and t in seconds, find the height of the tower.    122.5 m

DERIVATIVE AS A RATE MEASURE 

EXERCISE - C

1) Find the rate of change of the area of a circle of radius r, when the radius varies. 2πr

2) Find the rate of change of the whole surface of a cylinder of radius r and height h, when the radius varies .        (4πr+ 2πh)

3) An edge of a variable cube is increasing at the rate of 3 cm per second. How fast is the volume of the cube increasing when the edge is 10cm long ?    900 cm³/s

4) A sphericals soap bubbles is expanding so that its radius is increasing at the rate of 0.02 cm per second. At what rate is the surface area increasing when its radius is 4cm? (Take π= 3.14).      2.0096 cm²/s

5) The volume of a spherical balloon is increasing at the rate of 20 cm³/s. Find the rate of change of its surface area at the instant when its radius is 8cm.    5 cm²/sec

6) The volume of a cube is increasing at a constant rate. Show that the increase in surface area varies inversely as the length of the edge of the cube.     

7) A particle moves along the curve, 6y= (x³+2). Find the points on the curve at which the y-coordinates is changing 8 times as fast as the x-coordinate.      (4,11) And (-4, -31/3)

8) A ladder 5 m long, is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall, at the rate of 2 m/s. How fast is its height on the wall decreasing, when the foot of the ladder is 4m away from the wall ?      (-8/3) m/s

9) A conical vessel whose height is 10m and the radius of whose base is 5 m is being filled with water at the uniform rate of 1.5 cubic metres/minute. Find the rate at which the level of the water in the vessel is rising when the depth is 4m.     3/8π m/min

10) The radius of a cylinder is increasing at the rate of 2m/s and its altitude is decreasing at the rate of 3m/s.  Find the rate of change of volume when radius is 3 metres and the altitude 5 metres.        33π cu. m/s

11) A man 160cm tall, walks away from a source of light situated at the top of a pole 6m high, at the rate of 1.1 m/s. How fast is the length of his shadow increasing when he is 1 metre away from the pole ?    0.4 m/s

RAW- C

1) Find the rate of change of the volume of a sphere of radius r with respect to a change in the radius.     dV/dr = 4πr²

2) Find the rate of change of the volume of a cylinder of radius r and height h with respect to a change in the radius.       dV/dr = 2πrh

3) Find the rate of change of the curved surface of a cone of radius r and height h with respect to a change in the radius .       dS/dr = π(h²+ 2r²)/√(h²+ r²)

4) The side of a square is increasing at the rate of 0.2 cm/s. Find the rate of increase of the perimeter of the square.      dP/dt = 0.8 cm/s

5) The radius of a circle is increasing at the rate of 0.7 cm/s. What is the rate of increase of its circumference?    dC/dt = 4.4 cm/s

6) The radius of a circle is increasing uniformly at the rate of 0.3 cm per second. At what rate is the area increasing when the radius is 10cm ) (π=3.14).   dA/dt =18.84cm/s

7) The side of a squares sheet of metal is increasing at 3 cm per minute. At what rate is the area increasing when the side is 10cm long ?   dA/dt = 60cm²/min

8) The radius of a circular soap bubble is increasing at the rate of 0.2 cm/s. Find the rate of increase of increase of its surface area, when the radius is 7cm.     dS/dt = 35.2 cm²/s

9) The radius of an air bubble is increasing at the rate of 0.5 cm per second. At what rate is the volume of the bubble increasing when the radius is 1 centimetre ?   dV/dt = 6.28 cm³/s

10) The volume of a spherical balloon is increasing at the rate of 25 cubic centimetres per second. Find the rate of change of its surface at the instant when its radius is 5cm.      dS/dt = 6 cm²/s

11) A balloon which always remains spherical, is being inflated by pumping in 900 cubic centimetres of gas per second. Find the rate at which the radius of the balloon is increasing with the radius is 15cm. (π=22/7).     dr/dt = 0.32 cm/s

12) The surface area of a spherical bubble is increasing at 2 cm²/s. When the radius of the bubble is 6cm, at what rate is the volume of the bubble increasing ?   dV/dt = 6 cm³/s

13) The bottom of a rectangular swimming tank is 25m by 40m. What is pumped into the tank at the rate of 500 cubic metres per minute. Find the rate at which the level of the water in the tank is rising .      dh/dt = 0.5 m/min

14) A stone is dropped into a quite lake and waves move in circle at a speed of 3.5 cm per second. At the instant when the radius of the circular wave is 7.5cm, how fast is the enclosed area increasing. (π=22/7).       165 cm²/s

15) A man 2 meters high, walks at a uniform speed of 5km per hour away from a lamp post, 6 m high. Find the rate at which the length of his shadow increases.     2.5 km/h

16) An inverted cone has a depth of 40cm and a base of radius 5cm. Water is poured into it at a rate of 1.5 cubic centimetres per minute. Find the rate at which the level of water in the cone is rising when depth is 4cm.     1/10π  cm/s

17) Sand is pouring from a pipe at the rate of 18cm³/s. The failling sand forms a cone on the ground in such a way that the height of the cone is one-sixth of the radius of the base. How fast is the height of the sand cone increasing when its height is 3 cm ?       1/18π     cm/s

18) Water is dripping out from a conical funnel at a uniform rate of 4 cm³/s through a tiny hole at the vertex in the bottom. When the slant height of the water is 3 cm, find the rate of decrease of the slant height of the water, given that the vertical angle of the funnel is 120°.        32/27π     cm/s

19) From a Cylindrical drum containing oil and kept vertical, the oil is leacking at the rate of 16ml/s. If the radius of the drum is 7cm and its height is 60cm, find the rate at which the level of oil is changing when oil level is 18cm.     16/49π cm/s

20) A ladder 13m long is leaning against a wall. The bottom of the ladder is pulled along the ground, away from the wall , at the rate of 2m/s. How fast is its height on the wall decreasing when the foot of the ladder is 5m away from the wall ?    5/6π   cm/s

21) A man is moving away from a tower 40m high at a speed of 2m/s. Find the rate at which the angle of elevation of the top of the tower is changing, when he is at a distance of 30 metres from the foot of the tower. Assume that the eye level of the man is 1.6m from the ground.     0.032 radian/second 

22) Find an angle x, which increases twice as fastest as its sine.     π/3

23) The radius of a balloon is increasing at the rate of 10 cm/sec. At what rate is the surface area of the balloon increasing when the radius is 15 cm?    1200π cm²/sec

24) An edge a variable cube is increasing at the rate of 5 cm/sec. How fast is the volume of the cube increasing when the edge is 10cm long?      1500 cm³/sec

ERRORS AND APPROXIMATION 

1) Find the approximate value of the cube root of 127.       377/75

2) Find the approximate value of (0.007)¹⁾³.        23/120

3) Find the approximate value of square root of 401.      20.025

4) Find the approximate value of √0.037.          0.1925

5) Find the value of log₁₀10.1 , where log₁₀e= 0.4343.      1.004343

6) Find the approximate value of tan46,  if it is being given that 1°=0.01745 radians.      1.03490

7) if the radius of a circle increases from 5cm to 5.1 cm, find the increase in area.    π cm²

8) if y= (x⁴- 12) and if x changes from 2 to 1.99. What is the approximate change in y ?       -0.32

9) The time T of oscillation of a simple pendulum of length l is given by T= 2π √(l/g). Find the percentage error in T corresponding to an error of 2% in the value of l.     1%

10) if the error committed in measuring the radius of a circle be 0.01%,  find the corresponding error in calculating the area.       0.02%

11) If in a Triangle ABC , the side c and the angle C remain constant, while the remaining elements are changed slightly, show that da/cosA + db/cosB =0.

12) The area S of a triangle is calculated is measuring b, c and A. if there be an error δA in the measurement of A, show that the relative error in area is given by δS/S = cosA. δA.

RAW- 

1) (26)¹⁾³.      80/27

2) (123)¹⁾³.        373/75

3) (7.5)¹⁾³.         1.96

4)  (0.009)¹⁾³.      0.208

5)  √25.2.       5.02

6)  √0.163.       0.404

7)  (255)¹⁾⁴.       3.9961

8) (15)¹⁾⁴.     63/32

9) Find the approximate value of 1/(2.002)².    0.2495

10) Calculate the approximate value of log₁₀(4.04), it being given that log₁₀4= 0.6021 and log₁₀e= 0.4343.      0.606443

11) Find the value of logₑ10.02, it being given that logₑ10 = 2..3026.       2.3046

12) if y= sin x and x changes from π/2 to 22/14, what is the approximate change in y ?      No change 

13) A circular metal plate under heating so that its radius increases by 2%. Find the approximate increase in the area of the plate, if the radius of the plate before heating is 10 cm.       4π cm²

14) Find the approximate the value of cos 61°, it being given that sin 60°= 0.86603 and 1°= 0.01745 radians .      0.4849

15) if the length of a simple pendulum is decreased by 2%, find the percentage decrease in its period T, where T= 2π √(l/g).        1%

16) The pressure p and the volume V of a gas are connected by the relation, pt¹⁾⁴= k, where k is a constant. Find the percentage increase in the pressure corresponding to a diminution of 0.5% in the volume.     0.125%

17) The radius of a sphere shrinks from 10cm to 9.8cm.  Find the approximate the decrease in

a)  volume .      80π cu.cm

b) surface area.       16π cm²

18) if there is an error of 0.1% in the measurement of the radius of a sphere, find the approximately the percentage error in the calculation of the volume of the sphere.    0.3%

19) Show that the relative error in the volume of sphere, due to error in measuring the diameter, is 3 times the relative error in the diameter.       

MAXIMUM AND MINIMUM 

EXERCISE - A

1) Find all the points of local maxima and minimum and the corresponding maximum and minimum values of the function f(x)= -3x⁴/4 - 8x³ - 45x²/2 + 105.    0, (105), -5, (295/4), -3,(231/4)

2) Find the local maximum and the local minimum the function 

a) f(x)= sin2x, where 0< x < π.    π/4,(1), 3π/4,(-1)

b) f(x)= (sin2x - x), where -π/2≤ x ≤π/2.    Min:-π/6,(-(√3/2)+π/6) max: π/6,(√3/2-π/6)

3) Find the local maximum and the local minimum of the function

a)  f(x)= (sinx - cosx), where 0< x<π/4.    Max:3π/4,(√2), min:7π/4,(-√2

b) f(x)= (2cosx + x), where 0< x <π.       Max:π/6,(√3+π/6), min: 5π/6,(5π/6 -√3)

4) Find the point of local Maxima or local minimum the function

f(x)= (sin⁴x + cos⁴x) in 0< x <π/2.      Min: π/4,(1/2)

5) Find the local Maxima and local minimum and the corresponding local maximum and local minimum value of the following functions:

a) f(x)= x √(1- x), where x > 0.    Max: 2/3,(2/3√3

b) f(x)= x/{(x -1)(x -4)}, where 1< x < 4.      Max: 2,(-1)

6) Prove that the maximum value of (1/x)ˣ is e¹⁾ᵉ.

7) Find the point of the parabola y²= 2x which is closest to the point (1,4).     (2,2)

8) Show that the none of the following function has a maximum or minimum 

a) eˣ

b) logx

c) x³+ x²+ x +1.

9) Show that sinᵖθ cosʳθ attain a maximum, when θ = tan⁻¹√(p/r).

10) Find the maximum and minimum values of 

(3x⁴- 8x³+ 12x² - 48x + 25) on [0,3].        L Min:2, min.v in [0,3] is 39 at x=2, max.v: in [0,3] is 25 at x=0

11) Find the maximum and minimum value of (x + sin2x) in [0,2π].max.v:2π at 0, min.v: at2π/3 (4π-3√3)/6

12) Show that sinx (1- cosx), x ∈ [0,π] is maximum at x= π/3

RAW- A

1) Find the maximum or minimum values of the following:

a) (5x -1)²+ 4.     Min.v: 4 

b) -(x -3)²+9.     Max.v: 9

c) -|x +4|+ 6.        Max: 6

d) sin2x +5.        Max: 4, min: 6

e) |sin4x + 3|.      Mx: 4, min: 2

2) Find the points of local maximum or local minimum and the corresponding local maximum and the minimum value of each of the following functions.

a) f(x)= (x -3)⁴.           Mx is 0 at 3

b) f(x)= = x².       Min is 0 at x=0

c) f(x)= 2x³ - 21x²+ 36x -20.      Max is -3 at x=1; min is -128 at x=6

d) f(x)= x³- 6x²+ 9x +15.       Max is 19 at x=1, min is 15 at x=3

e) f(x)= x⁴- 62x²+ 120x +9.      Max is 68 at x=1, min are -1647 at x=-6 and -316 at x= 5

f) f(x)= - x³+ 12x²- 5.        Max is 251 at x=8 , min is -5 at x=0

g) f(x)= (x -1)(x +2)².     Max is 0 at x=-2, min is -4 at x=0

h) f(x)= -(x -1)³(x +1)².     Max is 0 at x=1 and x= -1 , min is -3456/3125 at x= -1/5

i) f(x)= x/2 + 2/x, x>0.     Min is 2 at x=2

 

3) Find the maximum and minimum values of (2x³- 24x +107) in the interval [-3,3].        Max is 139 at x=-2 , min is 189 at x=3

4) Find the maximum and minimum values of 3x⁴- 8x³+ 12x²- 48x +1 on the integral [1,4].           Max is 237 at x=4, min is -63 at x= 2

5) Find the maximum and the minimum values of

 f(x)= {sinx + (1/2) cosx} in 0≤ x ≤π/2.     Max is 3/4 at x=π/6, min is 1/2 at x= π/2

6) Show that the maximum value of x¹⁾ˣ is e¹⁾ˣ.

7) Show that (x + 1/x) has a maximum and minimum, but the maximum value is less than the minimum value.

8) Find the maximum profit that a company can make, if the profit function is given by p(x)= 41+ 24x - 18x².       49

9) An enemy jet is flying along the curve y= (x²+2). A soldier is placed at the point (3,2). Find the nearest point between the soldier and the jet.      (1,3)

10) Find the maximum and minimum values of f(x)= (-x + 2 sinx) in [0,2π].       Max is (-π/3+ √3) at x=π/3, min is -(5π/3+ √3) at x= 5π/3

EXERCISE - B

1) Amongst all pairs of positive numbers with sum 24, find these whose product is maximum.         12,12

2) Amongst all pairs of positive numbers with product 256, find those whose sum is the least.      16,16

3) Find two positive numbers X and y such that, (x + y)= 60 and xy³ is maximum.     45,15

4) Find two numbers whose sum is 16 and the sum of whose cubes is minimum.     8,8

5) Show that all the rectangles with a given perimeter, the square has the largest area.      

6) Show that, of all the rectangles of a given area, the square has the smallest perimeter.       

7) Prove that the area of a right angled triangle of a given hypotenuse is maximum when the triangle is isosceles.       

8) If the sum of the lengths of the hypotenuse and a side of a right angled triangle is given, show that area of the triangle is maximum when the angle between them is (π/3).

9) Two sides of a triangle are given. Find the angle between them such that the area shall be maximum.

10) Show that, of all the rectangle inscribed in a given fixed circle, the square has the maximum area.

11) Show that the triangle of maximum area that can be inscribed in a given circle is an equilateral triangle.

12)  The combined resistance R of a two resistors R₁ and R₂ where R₁, R₂ > 0  is given by 1/R = 1/R₁ + 1/R₂. If R₁+ R₂ = C(constant), show that the maximum resistance R is obtained by choosing R₁ = R₂.

13) A beam of length l is supported at one end. if W is the uniform load per unit length, the bending moment M at a distance x from the end is given by. M=(lx/2 - Wx²/2).  Find the point on the beam at which the bending moment has the maximum value.      1/2W

14) A wire of length 25 m is to be cut into two pieces. One of the wires is to be made into square and the other into a circle. What should be the length of the two pieces so that the combined area of the square and the circle is minimum ?     100/(π+4), 25π/(π+4)

15) Show that a cylinder of a given volume which is open at the top has minimum total surface area, provided its height is equal to the radius of its base.

16) Show that the height of a cylinder, which is open at the top, having a given surface and the greatest volume, is equal to the radius of its base.

17) Show that the semi vertical angle of a cone of maximum volume and of a given slant height is tan⁻¹√2.

18) Show that the semi vertical angle of a right circular cone of given surface area and maximum volume is sin⁻¹(1/3).

19) Show that the surface area of a closed cuboid with square base and given volume is minimum, when it is a cube.

20) Show that the height of a closed cylinder of given surface and maximum volume, is equal to the diameter of its base.

21) A closed right circular cylinder has volume 2156  cubic units. What should be the radius of its base, so that its total surface area may be maximum ?

22) Show that the height of the cylinder of maximum volume that can be inscribed in a sphere of radius R is 2R/√3. Find the volume of the largest cylinder inscribed in a sphere of radius R.    

23) Show that the cone of the greatest volume which can be inscribed in a given sphere is such that three times its altitude is twice the diameter of the sphere. Find the volume of the largest cone inscribed in a sphere of radius R.

24) prove that the parameter of a right angled triangle of given hypotenuse is maximum when the triangle is isosceles.

25) An open box is to be made out of a piece of cardboard measuring (24cm x  24 cm) by cutting off equal squares from the corners in the turning of the side. Find the height of the box having maximum volume.     4cm

BAYE'S THEOREM

EXERCISE A

1) In a bolt factory, machine A, B and C manufacture respectively 25%, 35% and 40% of the total bolts. Of their output 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. If the bolt drawn is found to be defective, what is the probability that is manufactured by the machine B? 28/69

2) Three urns contain 6 red, 4 Black: 4 red, 6 black; and 5red , five black balls respectively. One of the urns is selected at random and a ball is drawn from it. If the ball drawn is red, find the probability that it is drawn from the first urn. 2/5

3) A company has two plants to manufacture scooters. Plant I manufactures 70% of the scooters and Plant II manufacturers 30%. At plant I, 80% of the scooters are rated as of a standard quality and at Plant II, 90% of the scooters are rated as of standard quality. A scooter is choosen at random and is found to be standard quality. What is the probability that it has come from Plant II? 27/83

4) An Insurance company insured 2000 scooter drivers , 4000 car drivers and 6000 truck drivers . The probability of an accident involving a scooter driver, car driver and a truck driver are 0.01, 0.03 and 0.15 respectively. One of the insured person meets with an accident. What is the probability that he is a scooter driver? 1/52

5) Urn A contains 2 white, one Black and 3 red balls , urn B contains 3 white, 2 black and 4 red balls and urn C contains 4 white, 3 black balls and 2 Red balls . One urn is chosen at random and 2 balls are drawn at random from the urn. If the chosen happen to be red and black, what is the probability that both balls come from urn B ? 20/53

6) There are 3 bags, each containing 5 white balls and 3 black balls. Also there are two bags, each containing two White balls and 4 black balls . A white ball is drawn at random. Find the probability that this white ball is from a bag of the first group. 45/61

7) A card from a pack of 52 cards is lost. From the remaining cards of the pack, Two cards are drawn and are found to be hearts. Find the probability of the missing card to be a heart. 11/50

8) Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin 3 times and notes the number of heads. if she gets a 1, 2, 3 or 4, she tosses a coin once and notes whether a head or tail is obtained. if she obtained exactly one head; What is the probability that she threw a 1, 2, 3 or 4 with the die? 8/11

9) Given 3 identical boxes I, ii and III, each containing two coins. In box I both coins are gold coins, in box II both are Silver coins and in box III there is one gold and one silver coin. A person chooses a box at random and takes out a coin. If the coin is of gold, What is the probability that the other coin in the box is also a gold ? 2/3

10) A bag I contains three red and 4 black balls and bag II contains 4 red and 5 black balls . One ball is transferred from bag I to bag II and then a ball is drawn from bag II. The ball so drawn is found to be red in colour. Find the probability that the transferred ball is black. 16/31

11) Suppose that 5% of men and 0.25% of women have grey hair. A grey haired person is selected at random. What is the probability of this person being male ? Assume that there are equal number of males and females. 20/21

12) A bag contains 4 balls. Two balls are drawn at random and are found to be white. What is the probability that all balls are white ? 3/5

13) A bag contains 3 red and 7 black balls. Two balls are selected at random one-by- one without replacement. If the second selected ball happens to be red, what is the probability that the first selected ball is also red ? 2/9

14) Suppose that 6% of the people with blood group O are left handed and 10% of those who other blood groups are left handed. 30% of the people have blood group O. If a left handed person is selected at random, what is the probability that he/she will have blood group O ? 9/44

15) A man is known to speak truth 3 out of 4 times. He throws a die and reports that it is a six. Find the probability that it is actually a six. 3/8

16) In a test, an examine either guesses or copies or knows the answer to a multiple choice question with 4 choices. The probability that he makes a guess is 1/3 and the probability that he copies the answer is 1/6. The probability that his answer is correct, given that he copied it, is 1/8. Find the probability that he knew the answer to the question, given that he correctly answered it. 24/29

17) A letter is known to have come from TATANAGAR or CALCUTTA . On the envelop just two consecutive letters TA are visible. What is the probability that the letter has come from 

a) CALCUTTA. 4/11

b) TATANAGAR. 7/11

18) A doctor is to visit a patient. From the past experience, it is known that the probability that he will come by train, bus, scooter or by other means of transport are respectively 3/10, 1/5,1/10 and 2/5. The probability that he will be late 1/4,1/3 and 1/12, if he comes by train, bus and scooter respectively, but if he comes by other means of transport, then he will not be late, When he arrives, he is late. What is the probability that he comes by train . 1/2

19) Let d₁, d₂, d₃ be three mutually exclusive diseases . Let S={s₁, s₂, s₃,..., s₆} be the set of observable symptoms of these diseases . For example s₁ is the shortness of breath , s₂ is loss of weight, s₃ is fatigue . Suppose a random sample of 10000 patients contains 3200 patients with disease d₁, 3500 with disease d₂ and 3300 with disease d₃. Also, 3100 patients with disease d₁, 3300 with disease d₂ and 3000 with disease d₃ show the symptom S. Knowing that the patient has symptoms S, the doctor wishes to determine the patient 's illness. On the basis of this information, what should the doctor conclude ? P(E₃/A)< P(E₁/A)< P(E₂/A) i.e., P(E₂/A) is largest. This, the doctor should conclude that the patient is most likely to have disease d₂

20) Suppose that the realibilty of HIV test is specified as follows:

Of people having HIV, 90% of the test detect the disease but 10% go undetected, Of people free of HIV, 99% the test are judged HIV-ve but 1% are diagnosed as showing HIV+ve. From the large population of which only 0.1% have HIV, one person is selected at random, given the HIV test, and the pathologist reports him/ her as HIV+ve. What is the probability that the person actually has HIV ? 90/1089

21) Suppose you have two coins which appear identical in your pocket. You know that one is fair and one is 2-headed. If you take one out, toss it and get a head, what is the probably that it was a fair coin ? 1/3

22) Three bags contains a number of red and white balls are as follows:

Bag I :3 red balls; Bag II : 2 red balls and 1 white ball; bag III : 3 white balls.

The probably that bag i will be chosen and a ball is selected from it is i/6, i= 1, 2, 3. If a white ball is selected, what is the probability that it come from 

a) bag I. 2/11

b) bag II. 9/11

c) bag III.

23) A shopkeeper sells three types of seeds A₁, A₂ and A₃. They are sold as a mixture where the proportion are 4:4:2 respectively. The germination rates of the three types of seeds are 45%, 60%, and 35%. calculate the probability

a) that it will not germinate given that seed is of type A₃. 0.65

b) of a randomly chosen seed to germinate. 0.49

c) that it is of type A₂ given that a randomly chosen seed does not germinate. 16/51

RAW - A

PROBABILITY - B

1) SOME BASIC TERMS AND CONCEPTS 

  a) An Experiment:

An action or operation resulting in two or more outcomes is called an experiment.

 

   b) Sample Space:

The set of all possible possible all possible possible outcomes of an experiment is experiment is called the sample space, denoted by S. An element of S is called sample point.

  c) Event :

Any subset of sample space is an event.

  d) Simple Event:

An event is called simple event if it is a singleton subset of subset of sample space S.

  e) Compound Events:

It is the joint occurrence of two or more simple events.

  f) Equally Likely Events:

A number of simple events are said to be equally likely if there is no reason for one event to occur to occur occur in preference to any other event.

  g) Exhaustive Events:

All the possible outcomes taken together in which an experiment can result as said to be exhaustive or disjoint.

  h) Mutually Exclusive or Disjoint Events:

If two events cannot occur simultaneously, then they are mutually exclusive. If A and B are mutually exclusive then A∩B=∅

  i) Complement of an Event:

The complement of an event A, denoted by A' , A'' or Aᶜ is the set of all sample points of the space other than the sample points in A

2) MATHEMATICAL DEFINITION OF PROBABILITY

      Let the outcomes of an experiment consists of n-exhaustive mutually exclusive and equally likely cases. Then the sample spaces S has n sample points. If an event A consists of m sample points, (0 ≤ m ≤ n), then the probability of event A, denoted by P(A) is defined to be m/n i.e., P(A) = m/n.

  let S= a₁, a₂, .... aₙ be the sample space.

     a) P(S)= n/n= corresponding to the certain event.

     b) P(∅)= 0/n =0 corresponding to the null event ∅ or impossible event.

    c) If Aᵢ = {aᵢ}, i = 1, .... n then Aᵢ is the event corresponding to a single sample point aᵢ , then P(A₁) = 1/n

   d) 0 ≤ P(A) ≤ 1.

3) ODDS AGAINST AND ODDS IN FAVOUR OF AN EVENT:

   

      Let there be be e m + n equally likely, mutually exclusive and exhaustive cases out of which an event A can occur in m cases and does not occur in n cases. Then by definition of probability of occurrences = m/(m+n)

  The probability of non-occurance = n/(m+n)

  So, P(A): P(A')= m: n

 Thus, the odds in favour of occurrences of the event A are defined by m:n i.e., P(A) : P(A') ; and the odds against the occurrence of the event A are defined by n: m i.e., P(A') : P(A)

4) ADDITION THEOREM:

    a) A and B are any event in S, then P(AUB) = P(A)+ P(B) - P(A∩B)

  Since the probability of an event is a non negative number, it follows that 

P(A UB) ≤ P(A) + P(B)

For three events A, B and C in we have P(AUBUC)= P(A)+ P(B)+ P(C) - P(A∩B) - P(B∩C) - P(C∩A) + P(A∩B∩C).

General Form Of Addition Theorem -->

  For n events A₁, A₂, A₃.... Aₙ in S, we have P(A₁UA₂)U A U A ... Aₙ)

= ⁿᵣ₌₁∑ P(Aᵢ)= ᵣ₌₁ ∑P(Aᵢ ∩Aᵢ) + ᵢ₌ⱼ₌ₖ∑ P(Aᵢ∩Aⱼ ∩ Aₖ) ..... + (-1)ⁿ⁻¹ P(A₁∩ A₂∩A ₃ .......∩ Aₙ)

   b) If A and B are mutually exclusive, then P(A∩ B)= 0 so that P(AUB) = P(A)+ P(B)

5) MULTIPLICATION THEOREM:

    Independent event:

So, if A and B are two independent events that happening of B will have no effect on A.

      Difference Between Independent & Mutually Exclusive Event:

    (i) Mutually exclusiveness is used when events are taken from same experiment & Independence when events one takes from different experiment.

   (ii) Independent events are represented by word "and" but mutually exclusive events are represented by word "OR" 

    (a) When Events are Independent: 

P(A/B) = P(A) and P(B/A)= P(B), then

P(A ∩ B)= P(A) . P(B). OR

P(AB) = P(A) . P(B)

     

      (b) When Events Are Not Independent-->

    The probability of simultaneous happening of two events A and B is equal to the probability of A multiplied by the conditional probability of B with respect to A (or probability of B multiplied by the conditional probability of A with respect to B) i.e.,

 P(A ∩ B)= P(A) . P(B/A). OR

 P(B) . P(A/B).    

          OR

P(AB)= P(A).P(B/A). OR

P(B) . P(A/B)

   c) Probability Of At Least One Of The n Independent Events-->

             IF P₁ .P₂.P₃......Pₙ are the probabilities of n independent events A₁. A₂.A₃....Aₙ then the probability of happening of at least one of these event is

     1 - [(p₁)(1- p₂)....(1 - pₙ)]

P[A₁ +A₂+A₃+ .....+ Aₙ) = 1 - P(A'₁) . P( A'₂). P(A'₃)........P(A'ₙ) 

6) CONDITIONAL PROBABILITY: 

     Conditional Probability:

 If A and B are any events in S then the conditional probability of B relative to A is given by

  P(B/A)= P(B∩A)/P(A) , If P(A)≠0

EXERCISE - A

1) Find the probability of getting a head in a throw of a coin.       1/2

2) In a simultaneous throw of two dice, find the probability of getting a total of 7 ?      1/6

3) A coin is toss successively three times. Find the probability of getting exactly one head or two heads.       3/4

4) Three unbiased coins are tossed . What is the probability of getting 

a) all head.            1/8

b) Two heads.        3/8

c) one head.             3/8

d)  at least one head.    7/8

e) at least two heads.    1/2

5)  What is the probability that a number selected from the numbers 1, 2, 3,... 24, 25, is a prime number, when each of the given numbers is equally likely to be selected ?     9/25

6) Tickets numbered from 1 to 20 are mixed up together and then a ticket is drawn at random. What is the probability that the ticket has a number which is a multiple of 3 or 7 ?       2/5

7) One card is drawn from a back of 52 cards, each of the 52 cards being equally likely to be drawn . Find the probability that 

a) the card drawn is red.      1/2

b) the card drawn is a king.      1/13

c) the card drawn is red and a king.       1/26

d) the card drawn is either red or a king.       7/13

8) What is the probability that a leap year selected at random will contains 53 Sundays ?     2/7

9) A bag contains 9 black and 12 white balls. One ball is drawn at random. What is the probability that the ball drawn is black ?        3/7

10) A bag contains 8 red and 5 white balls . 3 balls are drawn at random . Find the probability that 

a) all the three balls are white.       5/143

b) all the three balls are red.         28/143

c) one ball is red and two ball are white .        40/143

11) Two cards are drawn at random from a pack of 52 cards.  What is the probability that the drawn cards are both access.         1/221

12) In a lottery 50 tickets numbered 1 to 50, two tickets are drawn simultaneously. Find the probability that 

a) Both the tickets drawn have prime numbers..   21/245

b) none of the tickets has prime number..       17/35

13) The odds in favour of an event are 3:5. Find the probability of occurrence of this event.          3/8

14) Three dice are thrown together. Find the probability of getting a total of atleast 6.             103/108

RAW- A

1) What is the probability of getting tails in a throw of a coin?      1/2

2) A die is thrown. Find the probability of 

a) getting 5.              1/6

b) getting 2 or 4.          1/3

c) getting an odd numbers.      1/2

d) getting a multiple of 3.      1/3

e) getting a prime number.      1/2

3) Two coins are tossed. Find the probability of 

a) getting 2 heads.       1/4

b) getting atleast one Heads.       3/4

c) getting no heads.      1/4

d) getting 1 head and 1 tails.        1/2

4) What is the probability that an ordinary year has 53 Sundays?     1/7

5) Tickets numbered from 1 to 12 mixed up together and then a ticket is drawn at random. Find the probability that the ticket has a number, which is multiple of 2 or 3.            2/3

6) A bag contains 4 white and 5 black balls. A ball is drawn at random from the bag. Find the probability that the ball is white.        4/9

7) In a single throw of two dice, find the probability of 

a) getting a total of 10.         1/12

b) getting a sum greater than 9.      1/6

c) getting a total of 9 or 11.       1/6

d) getting a doublet.      1/6

e) getting a doublet of even numbers.     1/12

8) Three coins are tossed. Find the probability of 

a) getting exactly 2 heads.      3/8

b) getting at least 2 heads.        1/2

c) getting at least one head and one tail.      3/4

9) In a lottery, there are 10 pricpzes ay25 blanks. Find the probability of getting a prize .       2/7

10) if there are two childrens in a family, find the probability that there is atleast one boy in the family.            3/4

11) Find the probability of getting a king or a queen, in a single draw, from a well shuffled pack of 52 cards.         2/13

12) An urn contains 9 red, 7 white and 4 black balls. A. Ball is drawn at random. Find the probability that the ball drawn is

a)  red.        9/20

b) white.     7/20 

c) red or White.      4/5

d) white or black.      11/20

e)  not red.        11/20

13) A card is drawn at random from a pack of 52 cards. Find the probability for drawing

a) a red card .          1/2

b) a black king.       1/26

c) a jack, a queen or a king.       3/13

d) either a red card or a king.        7/13

e) a card which is neither a heart nor a king.      9/13

14) In a single through of three dies, find the probability getting a total of 17 or 18.     1/54

15) Two dice are thrown. Find the probability that a multiple of 2 occurs on one die and a multiple of 3 occurs on the other.           11/36

16) A card is drawn at random from a well shuffled pack of 52 cards. Find the probability that it is neither an ace nor a king.       11/13

17) If odd in favour of an event be 2:3, find the probability of occurrence of this event.     2/5

18) If odd against an event be 3:4, find the probability of occurrence of this event.      4/7

19) a bag contains 4 white, 5 black and 7 red balls. 3 balls are drawn at random. Find the probability that all of them are black.      1/56

20) a bag contains 5 green and 7 red balls. Two balls are drawn at random. What is the probability that one is  green and the other red.      35/66

21) A bag contains 6 red, 4 White and 8 blue balls. If 3 balls are drawn at random, find the probability that 

a) one is red and 2 are white.         3/68

b) 2 are blue and one is red.       7/34

c) none is red.       55/204

22) A box contains four defective and 9 non-defective bulbs. Two bulbs are chosen at random. What is the chance that the both the bulbs are non defective.     6/13

23) A word consists of 9 letters; 5 consonants and 4 vowels . 3 letters are choose at random . What is the probability that more than one vowels will be selected.     17/42

24) A letter of the English alphabet is chosen at random . Calculate the probability that the letter so chosen.

a) is a vowel.        5/26

b) precedes and is a vowel.       3/26

c) follows k and is a vowel.         1/13

25) In a class there are 10 boys and 5 girls. Three students are selected at random. What is the probability that one girl and two boys are selected ?       45/91

26) Four persons are chosen at random from a group on containing three men, two women and 4 children. Find the chance that exactly two of them will be children.     10/21

27) From a group of 3 boys and two girls, two childrens of selected at random. Find the probability that at least one girl is selected.     7/10

28) A box contains 15 electric bulbs out of which 2 are defective. Two bulbs are chosen at random from the box. What is the probability that atleast one of these is defective.     9/35

29) In a single throw of two dice, find the probability of not getting the same number on both the dice .        5/6

30) Four cards are drawn at random from a pack of 52 cards. Find the probability of getting all the four cards of the same number.        1/20825

31) Two dice are thrown together. What is the probability that the sum of the numbers on the two faces is divisible by 3 or 4 ?       5/9

32) Find the probably getting the sum as a prime number when two dice are thrown together.        5/12

EXERCISE -B

1) Let A, B,C are arbitrary events. Find the expression for the events noted below, in the context of A, B and C.

a) Only A occurs.

b) Both A and B, but not C occur

c) All the three events occur 

d) Atleast one occurs

e) Atleast two occur.

f) One and no more occurs.

g) Two and no more occur.

h) None occurs.

i) Not more than two occurs.

2) An experiment involves rolling a pair of dice and recording the numbers that come up. Describe the following events.

A= the sum is greater than 8.

B= 2 occurs on either die.

C= the sum is at least 7 and a multiple of 3.

 Also, find A ∩ B, B ∩ C and A ∩ C.

Are: i) A and B mutually exclusive ?

       ii) B and C matually exclusive ?

      iii) A and C mutually exclusive ?

3) From a group of two boy and three girls, two children are selected at random. Describe the events .

i) A= both selected children are girls.

ii) B= the selected group consists of one boy and one girl.

iii) C= at least one boy selected.

Which pair/s of events is/are mutually exclusive?

4) Two dice are thrown and the sum of the numbers which come up on the dice is noted. Let us considered the following events.

A= the sum is even.

B= the sum is multiple of 3.

C= the sum is less than 4

D= the sum is greater than 11.

Which pairs of these events are mutually exclusive?

RAW - B

1) A coin is tossed. Find the total number of elementary events and also the total number events associated with the random experiment.

2) List all events associated with the random experiment of tossing of two coins. How many of them are elementary events.

3) 3 coins are tossed once. Describe the following events associated with this random experiment:

A= getting 3 heads

B= getting two heads and one tail

C= getting three tails 

D= getting a head on the first coin.

i) Which pairs of events are mutually exclusive?

ii) Which events are elementary events?

iii) Which events are compound events?

4) In a single throw of a die describe the following events:

i) A= Getting a number less than 7.

ii) B= Getting a number greater than 7.

iii) C= Getting a multiple of 3

iv) D= Getting a number less than 4

v) E= Getting an even number greater than 4 

vi) F= Getting a number not less than 3

 Also, Find A U B, A∩ B, B ∩ C, E ∩ F, D ∩ F and F'.

5) three coins are tossed. Describe 

i) two events A and B which are mutually exclusive.

ii) three events A, B and C which are mutually exclusive and exhaustive .

iii) Two events A and B which are not mutually exclusive.

iv) two events A and B which are mutually exclusive but not exhaustive .

6) A die is thrown twice. Each time the number approaching on it is recorded. Describe the following events:

i) A= both numbers are odd.

ii) B= both numbers are even.

ii) C= sum of the numbers is less than 6

Also, find A U B, A ∩ B, A U C , A∩ C

Which pairs of events are mutually exclusive?

7) Two dice are thrown. The events A, B, C, D, E and F are described as follows:

A= Getting an even number on the first die.

B= Getting an odd number on the first die.

C= Getting atmost 5 as sum of the numbers on the two dice.

D= Getting the sum of the numbers on the dice greater than 5 but not less than 10.

E= Getting at least 10 as the sum of the numbers on the dice.

F= Getting an odd number on one of the dice.

i) Describe the following events:

A and B, B or C, B and C, A and E, A or F, A and F

ii) State true or false:

a) A and B are mutually exclusive.

b) A and B are mutually exclusive and exhaustive events.

c) A and C are mutually exclusive events.

d) C and D are mutually exhaustive and exhaustive events.

e) C, D and E are mutually exclusive and exhaustive events.

f) A' and B' are mutually exclusive events.

g) A, B, F are mutually exclusive and exhaustive events.

8) The number 1, 2, 3 and 4 are written separately on four slips of paper. The slips are then put on a box and mixed thoroughly. A person draws two slips from the box, one after the other, without replacement. Describe the following events:

A= The number on the first slip is larger than the one of the second slip.

B= The number on the second slip is greater than ².

C= The sum of the numbers of the two slips is 6 or 7.

D= The number on the second slips is twice that on the first slip.

Which pairs/s of events is/are mutually exclusive

EXERCISE - C

1) Find the probability of getting a head in a toss of an unbiased coin. 

2) In a simultaneous toss of two coins, find the probability of getting:

a) two heads. 

b) exactly one head. 

c) exactly 2 tails. 

d) exactly one tail.

e) No tails.

3) 3 coins are tossed once . Find the probability of getting 

a) all heads. 

b) at least 2 heads. 

c) atmost two heads .

d) no heads.

e) exactly one tail. 

f) exactly two tails. 

g) a head on first coin. 

4) A dice is thrown. Find the probability of getting

a) an even number. 

b) a prime number. 

c) a number greater than or equals to 3. 

d) a number less than or equal to 4. 

e) a number more than 6. 

f) a number less than or equals to 6. 

5) Two dice are thrown simultaneously. Find the probability of getting:

a) an even number as the sum.

b) the sum as a prime number.

c) a total of at least 10. 

d) a doublet of even number. 

e) a multiple of 2 on one dice and a multiple of 3 on the other dice. 

f) same number on both dice. 

g) a multiple of 3 as the sum. 

6) Find the probability that a leap year, selected at random, well contains 53 Sundays. 

7) Three dice are thrown together. Find the probability of getting a total of atleast 6. 

8) What is the probability that a number selected from the number 1, 2, 3,....25, is prime number, when each of the given number is equally likely to be selected ? 

9) Tickets numbered from 1 to 20 are mixed up together and then a ticket is drawn at random. What is the probability that ticket has a number which is a multiple of 3 or 7 ? 

10) A coin is tossed . If head comes up, a die is thrown but if tails comes up, the coin is tossed again. Find the probability of obtaining :

a) two tails. 

b) head and number 6. 

c) head and an even number. 

11) A letter is chosen at random from the word ASSASSINATION. Find the probability that the letter is 

a) a vowel. 

b) a constant. 

12) In a lottery, a person choses six different natural numbers at random from 1 to 20, and if these six numbers match with the six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game? 

13) On her vacations Bina visits four cities A, B C and D in a random order. What is the probability that she visits.

a) A before B? 

b) A before B and B before C. 

c) A first and B last. 

d) A either first or second? 

e) A just before B? 

14) A die has two faces each with number '1', three faces each with number '2' and one face with number '3'. If die is rolled once determine:

a) P(2). 

b) P(1 or 3). 

c) P(not 3). 

15) If 4 digit numbers greater than or equals to 5000 are randomly formed from the digits 0, 1, 2, 3, 4 and 7, what is the probability of forming number divisible by 5 when

a) the digits may be repeated.

b) the repetition of digits not allowed. 

16) A fair coin is tossed four times, and a person wins Rs1 for each head and lose Rs1.50 for each tail that turns up. From the sample space calculate how many different amounts of money you can have after four tosses and probability having each of these amounts. 

17) One card is drawn from a pack of 52 cards , each the 52 cards being equally likely to be drawn. Find the probability that the card drawn is:

a) an ace. 

b) red. 1/2

c) either red or king. 

d) red and a king. 

18) An urn contains 9 red, 7 white and four black balls. If two balls are drawn at random, find the probability that:

a) both the balls are red. 

b) one ball is white.

c) the balls are of the same colour.

d) one is white and other red. 

19) A box contains 10 red marbles, 20 blue marbles and 30 green marbles, 5 marbles are drawn from the box, what is the probability that

a) all will be blue. 

b) atleast one will be green . 

20) In a lottery 10000 tickets are sold and ten equal prizes are awarded. What is the probability of not getting a prize if you buy 

a) a ticket.

b) two tickets. 

c) 10 tickets. 

21) The number lock of a suitcase has 4 wheels, each labelled with 10 digits i.e., from 0 to 9. The lock opens with a sequence of four digits with no repeats. What is the probability of a person getting the right sequence to open the suitcase. 

22) 3 letters are dictated to 3 persons on an envelope is addressed to each of them, the letters are inserted into the envelope at random so that each envelope contains exactly one letter, find the probability that at least one letter is in its proper envelope. 

23) Out of 100 students, two sections of 40 and 60 students are formed . if you and our friends are among the hundred students, what is the probability that 

a) you both enter the same section ? 

b) you both enter the different sections? 

24) Four cards are drawn at random from a pack of 52 playing cards. Find the probability of getting.

a) all the foure cards of the same suit. 

b) all the four cards of the same number. 

c) one card from each suit. 

d) two cards and two black cards. 

e) all cards of the same colour.

f) all face cards. 

25) In a lottery of 50 tickets numbered 1 to 59. Two tickets are drawn simultaneously. Find the probability that:

a) both the tickets drawn have prime numbers. 

b) none of the tickets drawn has prime number.

c) One ticket has prime number. 

26) A word consists of 9 letters; 5 constants and 4 vowels. Three letters are chosen at random. What is the probability that more than one vowel will be selected? 

27) A bag contains 50 tickets numbered 1,2,3,....,50 of which five are drawn at random and arranged in ascending order of magnitude (x₁ < x₂< x₃< x₄< x₅). Find the probability that x₃= 30. 

28) Four persons are to be chosen at random from a group of 3 men, 2 women and 4 children. Find the probability of selecting:

a) one man, one woman and two children. 

b) exactly two children. 

c) 2 women. 

29) A box contains 10 bulbs, of which just three are defective. If a random sample of five bulbs is drawn, find the probability that sample contains:

a) exactly one defective bulb .

b) exactly two defective bulbs.

c) no defective bulbs.

30) A bag contains tickets numbered 1 to 30. Three tickets are drawn at random from the bag. What is the probability that the maximum number of the selected tickets exceeds 25.

31) 12 balls are distributed among three boxes, find the probability that the first box will contain three balls. 

32) Find the probability that the birth days of six different persons will fall in exactly two calendar months. 

33) five marbles are wrong from a bag which contains 7 blue marbles and 4 black marbles. What is the probability that 

a) all will blue? 

b) 3 will be blue and 2 black? 

34) Find the probability that when a hand of seven cards is dealt from a well shuffled pack of 52 cards, it contains.

a) All 4 Kings.

b) exactly 3 kings. 

c) at least 3 Kings. 

35) In a single throw 3 dice, determine the probability of getting

a) a total of 5. 

b) a total of atmost 5. 

c) a total of atleast 5. 

36) Three dies are thrown simultaneously. Find the probability that:

a) all of them show the same face. 1/36

b) all show distinct faces.

c) two of them show the same face. 

37) What is the probability that in a group of 

a) two people, both will have the same birthday ? 

b) three people, at least two will have the same birthday ? 

38) The letters of word SOCIETY are placed at random in a row. What is the probability that three vowels come together.

39) Find the probability that in a random arrangement of the letters of the word UNIVERSITY the two 'I's come together. 

40) Five persons entered the left cabin on the ground floor of an an 8-floor house. Suppose that each of them independently and with equal probability can leave the cabin at any floor beginning with the first. Find out the probability of all five persons leaving at different floors. 

41) If n persons are selected on a random table, what is the probability that two named individuals will be neighbours ? 

42) A five digit number is formed by the digit 1, 2, 3, 4, 5 without repetition. Find the probability that the number is divisible by 4.

43) Out of 9 outstanding students in a college, there are four boys and 5 girls. A team of four students is to be selected from a quiz programme . Find the probability that two are boys and two are girls. 

44) In a lot of 12 microwave ovens , there are three defective units. A person has ordered 4 of these units and since each is identically packed, The selection will be random. What is the probability that

a) all four units are good. 

b) exactly 3 units are good. 

c) at least two units are good.

45) There are four letters and 4 addressed envelopes. Find the probability that all the letters are not dispatched in right envelopes.

46) The odds in favour of an event are 3:5. Find the probability of occurrence of this event. 

47) A fair coin with 1 marked on one face and 6 on the other and a fair die are both tossed, find the probability that sum of numbers that turns up is

a) 3. 

b) 12.

48) In a relay race that are 5 teams A, B , C and D and E.

a) What is the probability that A, B and C finish first, 2nd and 3rd respectively.

b) What is the probability that A, B and C are first three to finish (in any order).

49) A card is drawn from an ordinary pack of 52 cards and a Gambler bets that, it is a spade or an ace. What are the odds against his winning this between ? 

RAW - C

1) A die is thrown. Find the probability of getting 

a) a prime number.

b) two or four. 

c) multiple of 2 or 3. 

2) In a simultaneous throw of a pair of dice, find the probability of a getting:

a) 8 as the sum. 

b) a doublet .

c) a doublet of prime numbers.

d) a doublet of odd numbers.

e) a sum greater than 9.

f) an even number on first. 

g) an even number on one and a multiple of 3 on the other. 

h) neither 9 nor 11 as the sum of the numbers on the faces.

i) a sum less than 6. 

j) a sum of less than 7. 

k) a sum more than 7. 

l) neither a doublet nor a total of 10.

m) odd number on the first and 6 on the second. 

n) a number greater than 4 on each die.

o) a total of 9 or 11.

p) a total greater than 8. 

3) In a single throw of three dies, find the probability of getting a total of 17 or 18. 

4) 3 coins are tossed together. Find the probability of getting:

a) exactly two heads. 

b) at least two heads. 

c) at least one head and one tail. 

5) What is the probability that an ordinary year has 53 Sundays ? 

6) What is the probability that a leap year has 53 Sundays and 53 Mondays ? 

7) A and B throw a pair of dice. if A throws 9, find B's chance of throwing a higher number.

8) In a single throw 3 dies, find the probability of getting the same number on all the three dies. 

9) two unbiased dice are thrown. Find the probability that the total of the numbers on the dice is greater than 10. 

10) A card drawn at random from a pack of 52 cards. Find the probability that the card is drawn is:

a) a black King.

b) either a black card or a king.

c) black and a king. 

d) a jack , queen or a King. 

e) neither a heart nor a king. 

f) spade or an ace. 

g) neither an ace nor a king. 

h) a diamond card. 

i) not a diamond card. 

j) a black card.

k) not an ace. 

l) not a black card.

11) In shuffling a pack of 52 cards, 4 are accidentally dropped; find the chance that the missing cards should be one from each suit.

12) From a deck of 52 cards, 4 cards are drawn simultaneously , find the chance that they will be four honours of the same suit. 

13) Tickets numbered from 1 to 20 are mixed up together and then a ticket is drawn at the random. What is the probability that the ticket has the number which is a multiple of 3 or 7 ? 

14) A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability that one is red, one is white and one is blue.

15) A bag contains 7 white , 5 black and 4 red ball. If 2 balls are drawn at random, find the probability that:

a) both the balls are white. 

b) one ball is black and the other red. 

c) both the balls are of the same colour. 

16) A bag contains 6 red, 4 white and 8 Blue balls. If 3 balls are drawn at random, find the probability that:

a) one is red and two are white. 

b) two are blue and one is red. 

c) one is red. 

17) Five cards are drawn from a pack of 52 cards. What is the chance that these 5 will contain:

a) just one ace.

b) at least one ace.

18) The face cards are removed from a full pack. Out of the remaining 40 cards, 4 are drawn at random. What is the probability that they belong to different suits ? 

19) In a hand at Whist, what is the probability that four Kings are held by a specified player ? 

20) A box contains 100 bulbs , 20 of which are defective. 10 bulbs are selected for inspection. Find the probability that:

a) all 10 are defective. 

b) all 10 are good. 

c) at least one is defective.

d) none is defective. 

21) Find the probability that in a random arrangement of the letyers of the word SOCIAL vowels come together. 

22) The letters of the word CLIFTON are placed at random in a row. What is the chance that the two vowels come together. 

23) The letter of the word FORTUNATES are arranged at random in a row. What is the chance that the two T come together. 

24) Find the probability that in a random arrangement of the letters of the word UNIVERSITY, the two I's do not come together. 

25) If odds in favour of an event be 2:3, find the probability of occurrence of this event.

26) If odds against an event 7:9, find the probability of non-occurence of this event. 

27) Two balls are drawn at random from a bag contains 2 white, 3 red, 5 green and 4 black balls, one by one without replacement. Find the probability that both the balls are of different colours. 

28) Two unbiased dice are thrown. Find the probability that:

a) neither a doublet nor a total of 8 will appear. 

b) the sum of the numbers obtained on the two dice is neither a multiple of 2 nor a multiple of 3. 

29) A bag contains 8 red, 3 white and 9 blue balls. If 3 balls are drawn at random, determine the probability that

a) all the three balls are Blue balls. 

b) all the balls are of different colours. 

30) A bag contains 5 red, 6 white and 7 black balls. Two balls are drawn at random. What is the probability that both balls are red or both are black. 

31) if a letter is chosen at random from English alphabet, find the probability that the letter 

a) a vowel. 

b) A consonant. 

32) In a lottery a person chooses six different numbers at random from 1 to 20, and if these six numbers match with 6 numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game ? 

33) 20 cards and numbered from 1 to 20. One card is drawn at random. What is the probability that the number on the cards is:

a) a multiple of 4.

b) not a multiple of 4 ?

c) odd ? 

d) greater than 12 ?

e) divisible by 5 ? 

f) not a multiple of 6 ? 

34) Two dice are thrown . Find the odds in favour of getting the sum

a) 4. 

b) 5. 

c) what are the odds against getting the sum 6 ? 

35) What are the odds in favour of getting a spade if the card drawn from a well shuffled pack of 52 cards ? What are the odds in favour of getting a king ? 

36) A box contains 10 red marbles, 20 blue marbles and 30 green marbles. 5 marbles are drawn at random. From the box, what is the probability that:

a) all are blue ?

b) at least one is green ? 

37) A box contains 6 red marbles numbered 1 through 6 and 4 white marbles numbered from 12 through 15. Find the probability that a marble drawn is:

a) white. 

b) white and odd numbered. 

c) even numbered. 

d) red or even numbered. 

38) A class consists of 10 boys and 8 girls. 3 students are selected at random. What is the probability that the selected group has 

a) all boys ? 

b) all girls ? 

c) one boy and two girls ? 

d) at least one girl ? 

e) at most one girl ? 

39) Five girls are drawn from a well shuffle pack of 52 cards. Find the probability that all the five cards are hearts. 

40) a bag contains tickets numbered from 1 to 20. Two tickets are drawn. Find the probability that 

a) both the tickets have prime numbers on them. 

b) on one there is a prime number and on the other there is a multiple of 4. 

41) An urn contains 7 white, 5 black and 3 red balls. Two balls are drawn at random. Find the probability that 

a) both the balls are red. 

b) one ball is red and the other is black. 

c) one ball is white. 

42) A committee of two persons is selected from two men and two women. What is the probability that the committee will have 

a) no man. 

b) one man. 

c) two men?         

43) Two are four men and six women on the city councils. If one council member is selected for a committee at random, how likely is that it is a women? 

ADDITION THEOREM ON PROBABILITY 

EXERCISE - A

1) Given P(A)= 3/5 and P(B)= 1/5, find P(A or B), if A and B are mutually exclusive events . 

2) A and B are two mutually exclusive events of an experiment. If P(not A)= 0.65 , P(A U B)= 0.65 and P(B)= p, find the value of p. 

3) A and B are two non-mutually exclusive events. If P(A)=1/4, P(B)= 2/5 and P(A U B)= 1/2, find the value of P(A∩B) and P(A∩B'). 3/20, 

Y to

4) If E and F are two events such that P(E)= 1/4, P(F)= 1/2 and P(E and F)= 1/8, find

a) P(E or F).

b) P(not E and not F).

5) The probability that atleast one of the event A and B occurs is 0.6. If A and B occurs simultaneously with probability 0.2, then find P(A')+ P(B').

6) Check whether the following probabilities P(A) and P(B) are consistently defined:

a) P(A)= 0.5, P(B)= 0.7, P(A∩B)= 0.6.

b) P(A)= 0.5, P(B)= 0.4, P(A U B)= 0.8. 

7) Events E and F are such that P(not E or not F)= 0.25. state whether E and F are mutually exclusive.   

8) A, B, C are 3 mutually exclusive when exhaustive events associated with random experiment. Find P(A), it being given that P(B)= 3/2 P(A) and P(C)= 1/2 P(B).      

9) A, B, C are event such that P(A)0.3. P(B)= 0.4, P(C)= 0.8, P(A∩B)= 0.08, P(A∩ C)=0.28,  P(A∩B∩C)= 0.09. If P(AUBUC)≥ 0.75, then show that P(B∩C) lies in the interval (0.23, 0.48).

10) The probability of two events A and B are 0.25 and 0.50 respectively. The probability of their simultaneous occurrence is 0.14. Find the probability that neither A nor B occurs.     

11) Find the probability of getting an even number on the first die or a total of 8 in a single throw of two dice.       

12) A die is thrown twice. What is the probability that atleast one of the two throws comes up with the number 4?      

13) One number is chosen from numbers 1 to 200. Find the probability that it is divisible by 4 or 6 ?       

14) A card is drawn from a deck of 52 cards. Find the probability of getting a king or a heart or a red card.    

15) A drawer contains 50 bolts and 150 nuts . Half of the bolts and half of the nuts are rusted. If one item is chosen at random, what is the probability that it is rusted or a bolt?       

16) Four cards are drawn at a time from a pack of 52 playing cards. Find the probability of getting all the four cards of the same suit.       

17) An integer is chosen at random from the numbers ranging from 1 to 50. What is the probability that the integer chosen is a multiple of 2 or 3 or 10 ?      

18) In an essay competition p, the odds in favour of competetors P,Q,R,S are 1:2, 1:3, 1:4 and 1:5 respectively. Find the probability that one of them wins the competition.    

19) Two dice are thrown together. What is the probability that the sum of the numbers on the two faces is neither divisible by 3 nor 4 ?        

20) Two circles dice are thrown together. What is the probability that the sum of the number on the two faces is divisible by 3 or 4 ?      

21)  2 cards are drawn from a pack of 52 cards. What is the probability that either both are red or both are kings.     

22) A basket contains 20 apples and 10 oranges out of which five apples and 3 oranges are defectivd. If a person takes out 2 at random what is the probability that either both are apples or both are good ?         

23) trhe probability that a person will get an electric contract is 2/5 and the probability that he will not get plumbing contract is 4/7. If the probability of getting at least one contract is 2/3. What is the probability that he will get both?      

24) A die has 2 faces each with number '1', three faces each with number '2' and 1 face with number '3'. if the die is rolled once, determine 

a) P(1).      

b) P(1 or 3).       

c) P(not 3).     

25) The probability that a student will receive A, B and C or D grade are 0.40, 0.35, 0.15, 0.10 respectively. Find the probability that a student will receive 

a) B or C grade.     

b)  atmost of C grade.       

26)  Let A, B, C be three events. If the probability of occurring exactly one event out of A and B is 1- x, out of B and C is 1- 2x, out of C and A is 1- x, and that of occuring three events simultaneously is x⅖, then show that the probability that atleast one out of A,B,C will occur is greater 1/2

27) For the three events A, B and C, P( exactly one of the events A or B occurs)= P(exactly one of the events B or C occurs)= P(exactly one of the events C and A occurs)= p and P(all the three events occur simultaneously)= p², where 0 < p <1/2. Then, find the probability of occurrence of atleast one of the three events A, B and C.      

28) For a post three persons A, B and C appear in the interview. The probability of A being selected is twice that B and the probability of B being selected is thrice that of C. What are the individual probabilities of A, B, C being selected?      

29) P and Q are two candidate seeking admission in IIT. The probability that P is selected is 0.5 and the probability that both P and Q are selected is at most 0.3. Prove that the probability of Q being selected is atmost 0.8.      

30) A box contains 6 red, 4 white and 5 black balls. A person draws 4 balls from the box at random. Find the probability that among the balls drawn there is at least one ball of each colour.      

31) The probability that a patient visiting a dentist will have a tooth extracted is 0.06, the probability that he will a cavity filled is 0.2 and the probability that he will have a tooth extracted as well as cavity filled is 0.03. What is the probability that a patient has either a tooth extracted or a cavity filled ?       

32) The probability that a person visiting a dentist will have his teeth cleaned is 0.44, the probability that he will have a cavity filled is 0.24. The probability that he will have his teeth cleaned or a cavity filled is 0.60. what is the probability that a person visiting a dentist will have his teeth cleaned and cavity filled?    

33) Probability that Nandu passes in mathematics is 2/3 and the probability that he passes in English is 4/9. If the probability of passing both courses is 1/4, what is the probability that Nandu will pass in atleast one of these subjects?     

34) Find the probability of atmost two tails or at least two heads in a toss of 3 coins.   

35) In a town of 6000 people 1200 are over 50 years old and 2000 are female. It is known the 30% of the females are over 50 years. What is the probability that a random chosen individual from the town either female or over 50 years ?      

36) frrom the employees of a company, 5 persons are selected to represent them in the managing committee of the company. Particulars of the 5 persons are as follows :

S.No:   person   Age(in years)

1          male        30

2          male        33

3          male        46

4          female     28

5          male         41

A person is selected at random from this group to act as a spokespersons. What is the probability that a spokesperson will be either male or over 35 years.  

37) Two students Anil and Seema appeared in an examination. The probability that Anil will qualify the examination is 0.05 and that Seema will qualify the examination is 0.10. The probability that both will qualify the examination 0.02. Find the probability that :

a) both Anil and Seema will not qualify the exam.      

b) atleast only one of them will not qualify the exam.   

c) only one of them will qualify the exam.     

38) In a class XI of a school 40% of the students study Mathematics and 30% study Biology. 10% of the class study both mathematics and biology. If a student is selected at random from the class, find the probability that he will be studying mathematics or biology or both.       

39) In a class of 60 students 30 opted for NCC, 32 opted for NSS and 24 opted for both NCC and NSS. If one of these students is selected at random. Find the probability that :

a) the student opted for NCC or NSS.       

b) the student has opted neither NCC or no NSS.       

c) the student has opted NSS but not NCC.      2

RAW-A

1) If A and B are two events associated with a random experiment such that P(A)= 0.3, P(B)= 0.4 and P(AUB)= 0.5, find P(A ∩B).       

2) if A and B are two events associated with a random experiment such that P(A)= 0.5, P(B)= 0.3 and P(A∩B)=  0.2, find P(AUB).    

3) If A and B are two events associated with a random experiment such that P(AUB)= 0.8, P(A∩B)= 0.3 and P(A')= 0.5, find P(B).      

4) Given two mutually exclusive events A and B such that P(A)= 1/2 and P(B)= 1/3, find P(A or B).        

5) if A and B be mutually exclusive events associated with a random experiment such that P(A)= 0.4 and P(B)= 0.5, then find 

a) P(AUB).      

b) P(A' ∩B').     

c) P(A' ∩B).     

d) P(A∩  B').     

6) A and B are two events such that P(A)= 0.54, P(B)= 0.69 and P(A∩B)= 0.35. find

a) P(AUB).      

b) P(A' ∩B').      

c) P(A ∩B').     

d) P(B ∩ A').    

7) Fill in the blanks in the following table:

     P(A)     P(B)    P(A∩B)   P(AUB)

a)  1/3       1/5      1/15        ____

b) 0.35      ___        0.25         0.6

c) 0.5        0.35      ___           0.7             

8) There are three events A, B, C one of which must and only one can happen, the odds are 8 to 3 against A. 5 to 2 against B, find the odds against C.      

9) One of the two events must happen. Given that the chance of one is two-third of the other, find the odds in favour of the other.       

10) A card is drawn at random from a well shuffled pack of 52 cards. Find the probability of its being a spade or a king.      

11) In a single throw of two dice, find the probability that neither a doublet nor a total of 9 will appear.        

12) A natural number is chosen at random from amongst first 500. What is the probability that the number so chosen is divisible by 3 or 5 ?    

13) A die is thrown twice. What is the probability that at least one of the two throws come up with the numbers 3 ?       

14) A card is drawn from a pack of 52 cards. Find the probability of getting an ace or a spade card.      

15) A card is drawn at random from a well shuffled deck of 52 cards. Find the probability of its being a spade or a king.

16) One number is chosen from numbers 1 to 100. Find the probability that it is divisible by 4 or 6 ?     

17) From a well shuffled deck of 52 cards. 4 cards are drawn at random. What is the probability that all the drawn cards are of the same colour .  

18) 100 students appeared for two examinations . 60 passed the first, 50 passed the second and 30 passed both. Find the probability that a student selected at random has passed at least one examination.      

19) a box contains 10 white, 6 red and 10 black balls. A ball is drawn at random from the box. What is the probability that the ball is drawn is either white or red?    

20) In a race, the odds in favour of horses A, B, C, D are 1:3, 1 : 4,  1:5 and 1:6 respectively. Find the probability that one of them wins the race.     

21) The probability that a person will travel by plane is 3/5 and that he will travel by train is 1/4. What is the probability that he/she will travel by plane or train.     

22) Two cards are drawn from a well shuffle pack of 52 cards. Find the probability that either both are black or both are kings .      

23) Two cards are drawn from a deck of 52 cards. What is the probability that two cards drawn are either aces or black cards ?    

24) a box contains 30 bolts and 40 nuts. Half of the bolts and half of the nuts are rusted. if two items are drawn at random, what is the probability that either both are rusted or both are bolts.       

25) An Integer is chosen at random from first 200 positive integers. Find the probability that the integer is divisible by 6 or 8.    

26) Find the probability of getting 2 or 3 tails when a coin is tossed four times.   

27) In an entrance test that is graded on the basis of two examinations , the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7. The probability of passing at least one of them is 0.95. What is the probability of passing both ?     

28) The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither 0.1. if the probability of passing the English examination is 0.75. What is the probability passing the Hindi examination ?   

RAW- B

1) A card is drawn at random from a well shuffled pack of 52 cards. Find the probability of its being a spade or a king. 

2) Two cards are drawn at random from a well shuffled pack of 52 cards. What is the probability that either both are red or both are Kings ? 

3) A natural number is chosen at random from amongst the first 500. What is the probability that the number so chosen is divisible by 3 or 5 ? 

4) Two dice are tossed once. Find the probability of getting an even number on first die, or a total of 8.

5) In a single throw of 2 dice, find the probability that neither a doublet nor a total of 9 will appear. 

6) A die is thrown twice. What is the probability that atleast one of the two throws comes up with the number 4 ? 

7) A card is drawn from a pack of 52 cards. Find the probability of a getting a king or a heart or a red card. 

8) The probabilities of occurrence of two events E and F are 0.25 and 0.5 0 respectively. The probability of their simultaneous occurrence 0.14. Find the probability that neither E occurs nor F occurs . 

9) The probability that at least one of the events A and B occurs 0.6. if A and B occur simultaneously with probability 0.2, Find P(A)' + P(B)'. 

10) In a given race, the odds in favour of horses A, B, C, D are 1:3,1:4,1:5, and 1:6 respectively. Find the probability that one of them wins the race. 

11) Four cards are drawn at a time from a pack of 52 playing cards. Find the probability of getting all the four cards of the same suit.

RAW- C

1) If A and B are two events associated with a random experiment such that P(A)= 0.35, P(A or B)= 0.85 and P(A and B)= 0.15. Find P(A). 

2) Two dice are tossed together. Find the probability of getting a doublet or a total of 6. 

3) In a single throw of two dice, find the probability that neither a doublet nor a total of 10 will appear. 

4) A natural number is choosen at random from among the first 500. What is the probability that the number so chosen is divisible by 3 or 5 ? 

5) A card is drawn at random from a well shuffled deck of 52 cards. Find the probability of its being a spade or a king.

6) Two cards are drawn at random from a well support pack of 52 cards. What is the probability that either both are red or both are Kings. 

7) A box contains 100 Bolt and 50 nuts . It is given that 50% bolts and 50% nuts are rusted. Two objects are selected from the boxe at random. Find the probability that either both are bolts or both are rusted. 

8) If A and B are two events such that P(A)= 0.5, P(B)= 0.3 and P(A and B)= 0.1 find 

a) P(A or B). 

b) P(A but not B). 

c) P(B but not A). 

d) P(neither A nor B). 

9) The probability that at least one of the events A and B occurs is 0.6. if the probability with simultaneous occurrence of A and B is 0.2. Find the probability of P(A')+ P(B'). 

10) The probabilities of the occurrence of two events A and B are 0.25 and 0.05 respectively . The probability of their simultaneous occurrence is 0.14. Find the probability that neither A nor B occurs . 

11) A card is drawn from a pack of 52 cards. Find the probability of getting a king or a heart or a red card.

RAW-D

1) If A and B are two events associated with a random experiment such that P(A)= 0.3, P(B)= 0.4 and P(AUB)= 0.5, find P(A ∩B).        0.2

2) if A and B are two events associated with a random experiment such that P(A)= 0.5, P(B)= 0.3 and P(A∩B)=  0.2, find P(AUB).     0.6

3) If A and B are two events associated with a random experiment such that P(AUB)= 0.8, P(A∩B)= 0.3 and P(A')= 0.5, find P(B).        0.6

4) Given two mutually exclusive events A and B such that P(A)= 1/2 and P(B)= 1/3, find P(A or B).        5/6

5) if A and B be mutually exclusive events associated with a random experiment such that P(A)= 0.4 and P(B)= 0.5, then find 

a) P(AUB).      0.9

b) P(A' ∩B').     0.1

c) P(A' ∩B).     0.5

d) P(A∩  B').     0.4

6) A and B are two events such that P(A)= 0.54, P(B)= 0.69 and P(A∩B)= 0.35. find

a) P(AUB).       0.88

b) P(A' ∩B').      0.12

c) P(A ∩B').     0.19

d) P(B ∩ A').     0.34

7) Fill in the blanks in the following table:

     P(A)     P(B)    P(A∩B)   P(AUB)

a)  1/3       1/5      1/15        ____

b) 0.35      ___        0.25         0.6

c) 0.5        0.35      ___           0.7             7/15, 0.5,0.15

8) There are three events A, B, C one of which must and only one can happen, the odds are 8 to 3 against A. 5 to 2 against B, find the odds against C.      43:34

9) One of the two events must happen. Given that the chance of one is two-third of the other, find the odds in favour of the other.        2:3

10) A card is drawn at random from a well shuffled pack of 52 cards. Find the probability of its being a spade or a king.       4/13

11) In a single throw of two dice, find the probability that neither a doublet nor a total of 9 will appear.        

12) A natural number is chosen at random from amongst first 500. What is the probability that the number so chosen is divisible by 3 or 5 ?      233/500

13) A die is thrown twice. What is the probability that at least one of the two throws come up with the numbers 3 ?       11/36

14) A card is drawn from a pack of 52 cards. Find the probability of getting an ace or a spade card.      4/13

15) A card is drawn at random from a well shuffled deck of 52 cards. Find the probability of its being a spade or a king.

16) One number is chosen from numbers 1 to 100. Find the probability that it is divisible by 4 or 6 ?      33/100

17) From a well shuffled deck of 52 cards. 4 cards are drawn at random. What is the probability that all the drawn cards are of the same colour .    92/883

18) 100 students appeared for two examinations . 60 passed the first, 50 passed the second and 30 passed both. Find the probability that a student selected at random has passed at least one examination.        4/5

19) a box contains 10 white, 6 red and 10 black balls. A ball is drawn at random from the box. What is the probability that the ball is drawn is either white or red?     8/13

20) In a race, the odds in favour of horses A, B, C, D are 1:3, 1 : 4,  1:5 and 1:6 respectively. Find the probability that one of them wins the race.     319/420

21) The probability that a person will travel by plane is 3/5 and that he will travel by train is 1/4. What is the probability that he/she will travel by plane or train.     17/20

22) Two cards are drawn from a well shuffle pack of 52 cards. Find the probability that either both are black or both are kings .      55/221

23) Two cards are drawn from a deck of 52 cards. What is the probability that two cards drawn are either aces or black cards ?     55/221

24) a box contains 30 bolts and 40 nuts. Half of the bolts and half of the nuts are rusted. if two items are drawn at random, what is the probability that either both are rusted or both are bolts.       185/483

25) An Integer is chosen at random from first 200 positive integers. Find the probability that the integer is divisible by 6 or 8.     1/4

26) Find the probability of getting 2 or 3 tails when a coin is tossed four times.    5/8

27) In an entrance test that is graded on the basis of two examinations , the probability of a randomly chosen student passing the first examination is 0.8 and the probability of passing the second examination is 0.7. The probability of passing at least one of them is 0.95. What is the probability of passing both ?     0.55

28) The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither 0.1. if the probability of passing the English examination is 0.75. What is the probability passing the Hindi examination ?   0.65

RAW- E

1) A card is drawn at random from a well shuffled pack of 52 cards. Find the probability of its being a spade or a king.          4/13

2) Two cards are drawn at random from a well shuffled pack of 52 cards. What is the probability that either both are red or both are Kings ?       55/221

3) A natural number is chosen at random from amongst the first 500. What is the probability that the number so chosen is divisible by 3 or 5 ?      233/500

4) Two dice are tossed once. Find the probability of getting an even number on first die, or a total of 8.         5/36

5) In a single throw of 2 dice, find the probability that neither a doublet nor a total of 9 will appear.     13/18

6) A die is thrown twice. What is the probability that atleast one of the two throws comes up with the number 4 ?            11/36

7) A card is drawn from a pack of 52 cards. Find the probability of a getting a king or a heart or a red card.          7/13

8) The probabilities of occurrence of two events E and F are 0.25 and 0.5 0 respectively. The probability of their simultaneous occurrence 0.14. Find the probability that neither E occurs nor F occurs .         0.39

9) The probability that at least one of the events A and B occurs 0.6. if A and B occur simultaneously with probability 0.2, Find P(A)' + P(B)'.        1.2

10) In a given race, the odds in favour of horses A, B, C, D are 1:3,1:4,1:5, and 1:6 respectively. Find the probability that one of them wins the race.    319/420

11) Four cards are drawn at a time from a pack of 52 playing cards. Find the probability of getting all the four cards of the same suit.         44/4165

RAW- F

1) If A and B are two events associated with a random experiment such that P(A)= 0.35, P(A or B)= 0.85 and P(A and B)= 0.15. Find P(A).       0.65

2) Two dice are tossed together. Find the probability of getting a doublet or a total of 6.    2/9

3) In a single throw of two dice, find the probability that neither a doublet nor a total of 10 will appear.      7/9

4) A natural number is choosen at random from among the first 500. What is the probability that the number so chosen is divisible by 3 or 5 ?      233/500

5) A card is drawn at random from a well shuffled deck of 52 cards. Find the probability of its being a spade or a king.       4/13

6) Two cards are drawn at random from a well support pack of 52 cards. What is the probability that either both are red or both are Kings.      55/221

7) A box contains 100 Bolt and 50 nuts . It is given that 50% bolts and 50% nuts are rusted. Two objects are selected from the boxe at random. Find the probability that either both are bolts or both are rusted.      0.58

8) If A and B are two events such that P(A)= 0.5, P(B)=  0.3 and P(A and B)= 0.1 find 

a) P(A or B).      0.7

b) P(A but not B).      0.4

c) P(B but not A).    0.2

d) P(neither A nor B).    0.3

9) The probability that at least one of the events A and B occurs is 0.6. if the probability with simultaneous occurrence of A and B is 0.2. Find the probability of P(A')+ P(B').   1.2

10) The probabilities of the occurrence of two events A and B are 0.25 and 0.05 respectively . The probability of their simultaneous occurrence is 0.14. Find the probability that neither A nor B occurs .     0.39

11) A card is drawn from a pack of 52 cards. Find the probability of getting a king or a heart or a red card.      7/13

RAW -G

1) If E and F be the events in a simple space such that P(E)= 0.5, P(F)= 0.3 and P(E∩ F)= 0.2, find P(E U F).       0.6

2) if E and F be the events in a sample space such that P(E)= 0.4, P(F)= 0.5 find P(E∩ F), if  P(E U F)= 0.6.          0.3

3) If E and F be the events in a sample space such that P(E U F)= 0.8,  P(E∩ F)= 0.3 and P(E)= 0.5, find P(F).      0.6

4) if E and F be mutually exclusive events in a sample space such that P(E)=0.4 and P(F)= 0.5, then find 

a) P(E UF).     0.9

b) P(E)'.       0.6

c) P(F)'.       0.5

d) P(E' ∩ F').      0.1

e) P(E' ∩ F).       0.5

f) P(E∩ F').      0.4

5) A and B are two mutually exclusive events of an experiment. If P(not A)= 0.65, P(A U B)= 0.65 and P(B)= p, find the value of p.         0.3

6) If E and F be events in a sample space such that P(E)= 0.3, P(F)= 0.2 and P(E∩ F) =0.1 find P(E' ∩ F) and P(E ∩ F').       0.1, 0.2

7) A, B and C are 3 mutually exclusive and exclusive events associated with a random experiment find P(A), it being given that P(B)= (3/2) P(A) and P(C)= (1/2) P(B).    4/13

8) The probability that a company executable will travel by plane 2/5 and that he will travel by train is 1/3. Find the probability of his travelling by plane or train .     11/15

9) One number is chosen from numbers 1 to 100. Find the probability that is divisible by 4 or 6 ?     33/100

10) From a well shuffled pack of cards, a card is drawn at random. Find the probability that it is either a queen or a heart.       4/13

11) From a well shuffled pack of 52 cards, a card is drawn to random. Find the probability that the drawn card is king or a queen.       2/13

12) a box contains 4 red balls, 5 green balls and 6 white balls. A ball is drawn at random from the box. What is the probability that the ball drawn is either red or green.      3/5

13) In a class of 200 student, 120 take mathematics, 110 take physics and 40 take both mathematics and physics. if a student is selected at random from this class, find the probability that the student chosen takes mathematics or physics.        19/20

14)  In a class, 30% of the students offered ward mathematics, 20% offered chemistry and 10% offered both. If a student is selected at random, find the probability that he has offered mathematics or chemistry.         2/5

15) A drawer contains 50 balls and 150 nuts. Half of the bolts and half of the nuts are rusted. If one item is chosen at random, what is the probability that it is rusted or a bolt.         5/8

16) A die is thrown twice. What is the probability that atleast one of the two numbers is 5?       11/36

17) From well shuffled pack by 52 cards, 4 cards are drawn at random. What is the probability that all the drawn cards are of the same colour.        92/1323

CONDITION PROBABILITY 

More on Condition Probability

1) If A and B are two events associated with a random experiment, then

* P(A ∩ B)= P(A) P(B/A), if P(A)≠0

OR

P(A ∩ B)= P(B) P(A/B), if P(B)≠0

* P(A/B)= P(A ∩ B)/P(B)

AND

P(B/A) = P(A ∩ B)/ P(A) 

2) If A is an event associated with the sample space S of a random experiment, then

 P(S/A)= P(A/A)= 1

3) Let A and B be two events associated with a random experiment and S be the sample space of C is an event such that P(C)≠ 0, then

P(A UB)/C)= P(A/C)+ P(B/C)- P((A ∩ B)/C)

In particular, if A and B are mutually exclusive events, then

 P((A U B)/C) = P(A/C) + P(B/C)

4) If A and B are two events associated with a random experiment, then

P(A'/B)= 1 - P(A/B)

5) P(A'∩C')= 1 - P(A U C)

                  =1 - {P(A) + P(C) - P(A∩C)}

6) P(A'/B')=P(A'∩B')/P(B')

7) P(B'/A')=P(A'∩B')/P(A')

                 

EXERCISE- A

1) A basket contents 20 apples and 10 oranges out of which 5 apples and 3 oranges are defective. If a person takes out 2 at random what is the probability that either both are apples or both are good ?     316/435

2) The probability that a person will get an electric contract is 2/5 and the probability that he will not get plumbing contract is 4/7. If the probability of getting at least one contract is 2/3, what is the probability that he will get both ?     17/105

3) The probability of simultaneous occurrence of at least one of two events A and B is p. If the probability that exactly one of A, B occurs is q, then show that P(A')+ P(B')= 2 - 2p + q.

4) Let A, B and C be three events . If the probability of occurring exactly one event out of A and B is 1 - x, out of B and C is 1 - 2x,  out of C and A is 1 - x, and that of occuring three events simultaneously is x²,  then prove that the probability that atleast one out of A, B, C will occur is greater than 1/ 2.

5) For the three events A, B and C, P( exactly one of the events A and B occurs)= P (exactly one of the events B or C occurs)= P (exactly one of the events C and A occurs) = p and P (all the three events occur simultaneously)= p², where 0 < p < 1/2. Then, find the probability occurrence at least one of the three events A, B, and C.    (3p+ 2p²)/2

6) The probabilities that a student passed in Mathematics , Physics and Chemistry are m, p and c respectively. (of the subjects , the students has a 75% chance of passing in at least one, a 50% chance of passing at least two and a 40% chance of passing in exactly two. Find the value of p + m + c.        27/20

7) Let there be a bag contains 5 white and 4 red balls. Two balls are drawn from the bag one after the other without replacement. Consider the following events:

A= drawing a white ball in the first draw,

B= drawing a red ball in the second draw.

8) Consider the random experiment of throwing a pair of dice and two events associated with it given by

A=  The sum of the numbers on two dice is 8

B= there is an even number on first die

9) A dice thrown twice and the sum of the members appearing is observed to be 6. What is the conditional probability that number 4 has appeared at least once ?  2/5

10) A fair dice is rolled. Consider the following events:

A: {1,3,5}, B={2,3}, and C={2,3,4,5}. Find 

a) P(A/B) and P(B/A).        1/2,1/3

b) P(A/C) and P(C/A).      1/2,2/3

c) P(A U B/C) and P(A ∩B/C).      3/4

d) P(A∩ B ∩C).        1/4

11) A coin is tossed 3 times. Find (E/F) in each of the following:

a) E= head on the third toss, F= heads on first two tosses.    1/2 

b) E= at least two heads, F= at most two heads.    3/7

c) E= at most two tails , F= at least one tail.       6/7

12) Two coins are tossed . Find P(E/F) in each of the following:

a) E: tail appears on one coins, F=  one coin shows head.     1

b) no tail appears, F= no head appears.         0

13) Mother, father and son line up at random for a family picture. Find P(AB), if A and B are defined as follows:

A= son on one end,

B= father in the middle .           1

14) A and B are two events such that P(A)≠ 0, find P(B/A), if

a) A is a subset of B.      1

b) A ∩B = ∅.       0

15) A couple has two children.  Find the probability that 

a) both the children are boys, if it is known that the older child is a boy.     1/2

b) Both childrens are girls, if it is known that older child is a girl.     1/2

c) both the children are boys, if it is known that at least one of the children is a boy.     1/3

16) A pair of dice is thrown. If the two numbers appearing on them are different, find the probability 

a) the sum of the number is 6.        2/15

b) the sum of the numbers is 4 or less.      1/15

c) the sum of the number is 4.      2/15

17) A die is thrown twice and the sum of the numbers appearing is observed to be 6. What is the probability that the number 4 has appeared at least once ?      2/5

18) A die is thrown three times. Events A and B are defined as below.

A=  getting 4 on the third die.

B=  getting 6 on the first and 5 on the second throw 

Find the probability of A given has already occurred.      1/6

19) A black and a red dice are rolled in order. Find the conditional probability of obtaining 

a) a sum greater than 9, given that the black die resulted in a 5.     1/3

b) a sum 8, given that the red die resulted in a number less than 4.      1/9

20) Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the dice again and if any other comes, toss a coin. Find the conditional probability of the event 'the coin shows a tail' given that 'at least one die shows a three.       0

21) In a school, there are thousand student, out of which 430 are girls. it is known that out of the 430, 10% ofthe girls study in class XII. What is the probability that a student chosen randomly studies in class XII given that the chosen student is a girl ?     1/10

22) An instructor has a question bank consisting of 300 easy True/false questions, 200 difficult true/false questions. 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be any easy question given that it is a multiple choice question ?     5/9

23) Given that A and B are two events such that P(A)= 0.6, P(B)= 0.3 and P( A∩ B)= 0.2, find P(A/B) and P(B/A).    2/3, 1/3

24) If P(A)= 6/11, P(B)= 5/11 and P(A U B)= 7/11 find 

a) P(A∩B).      4/11

b) P(A/B).     4/5

c) P(B/A).      2/3

25) Evaluate P(AUB), if 2P(A)= P(B)= 5/13 and P(A/B)= 2/5.     11/26

26) Two integers are selected at random from integers 1 and 11. If the sum is even, find the probability that both the numbers are odd.        3/5

27) A die is thrown three times, if the first throw is a four, find the chance of getting 15 as the sum.         1/18

RAW - A(1)

1) Ten cards numbered 1 through 10 are placed in a box, mixed up throughly and then one card is drawn randomly. If is known that the number on the drawn card is more than 3, what is the probability that it is an even number.       4/7

2) Assume that each child born is equally likely to be a boy or a girl. If a family has two children, what is the conditional probability that both are girls given that 

a) the youngest is a girl.     1/2

b) at least one is a girl ?      1/3

3) Given that the two numbers are appearing on throwing two dice are different. Find the probability of the event 'the sum of the numbers the dice is 4'.      1/15 

4) A coin is tossed 3 times, if head occurs on first two tosses, find the probability of getting head on third toss.       1/2

5) A die is thrown 3 times, find the probability that 4 appears on the third toss if it is given that 6 and 5 appear respectivelly on first two tosses.     1/6

6) Compute P(A/B), if P(B)= 0.5 and P(A∩B)= 0.32.       16/25

7) If P(A)= 0.4, P(B)= 0.3 and P(A/B)= 0.5, find P(A∩B) and P(A/B).     0.2, 2/3

8) If A and B are two events such that P(A)= 1/3, P(B)= 1/5 and P(AUB)= 11/30, find P(A/B), P(B/A).        5/6, 1/2

9) A couple has two children. Find the probability that both children are 

a) males, if it is known that atleast one of the children is male.      1/3

b) females, if it is known that the elder child is a female.       1/2

RAW- A(2)

1) Let E and F be the event such that P(A)= 1/4 and P(A∩B)= 1/5, find 

a) P(E/F).       4/5

b) P(F/E).    3/5

c) P(EUF).       23/60

d) P(F'/E').      37/40

2) If A and B are two events such that P(A)= 0.4, P(B)= 0.8 and P(B/A)=  0.6,  find P(A/B), and P(AU B).        0.3, 0.96

3) ay die rolled, if the outcome is an odd number, what is the probability that it is prime?    2/3

4) Three fair coins of tossed. Find the probability that they are all tails, if one of the coins shows a tail.          1/7

5) a coin is tossed twice and the four possible outcomes are assumed to be equally likely. if E is the event, 'both head and tail have appeared', and F be the event, 'atmost one tail is observed',  find P(E), P(F), P(E/F), P(F/E).       1/2, 3/4, 2/3, 1

6) A dice thrown twice and the sum of the numbers appearing is observed to be 7. What is the conditional probability that the number 2 has atleast once?     1/3

7) Two unbiased dice are thrown. Find the probability that the sum is 8 or greater, if 4 appears on the first time die.        1/2

8) In a class, 40% students read mathematics, 25% biology and 15% both Mathematics and biology. One student is selected at a random. Find the probability that 

a) he reads mathematics, if it is known that he reads biology.     3/5

b) he reads biology, if it is known that he reads mathematics.     3/8

9) A couple has two children. Find the probability that a both are boys, if it is known that

a) one of the children is boy.      1/3

b) the older child is a boy.      1/2

10) Two numbers are selected at random from the integers 1 through 9. If the sum is even, find the probability that a both numbers are odd.        5/8

MULTIPLICATION THEOREM ON PROBABILITY 

EXERCISE - F

1) A bag contains 10 white and 15 black balls. Two balls are drawn in succession without replacement. What is the probability the first is white and second is black ?     1/4

2) Find the probability of a drawing a diamond card in each of the two consecutive draws  from a well shuffled pack of cards , if the card is drawn is not replaced after the first draw .        1/17

3) A bag contains 5 white, 7 red and 8 black balls. If four balls are drawn one by one without replacement, find the probability of getting all white balls .   1/969

4) A bag contains 19 tickets from 1 to 19. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will shown even numbers.        4/19

5) An urn contains 5 white and 8 black balls. Two successive drawings of three balls at a time are marked such that the balls are not replaced before the second draw. Find the probability that the first draw gives 3 white balls and second draw gives 3 black balls.      7/429

6) Two balls are drawn from an urn containing 2 white, 3 red and 4 black balls one by one without replacement. What is the probability that atleast one ball is red?     7/12

7) To test the quality of electric bulbs produced in a factory, two bulbs are randomly selected from a large sample without replacement. If either bulb is defective, the entire lot is rejected. Suppose a sample of 200 bulbs contains 5 defective bulbs. Find the probability that the sample will be rejected.      197/3980

8) A bag contains n white and n red balls. Pairs of balls are drawn without replacement until the bag is empty. Show that the probability that each pair consists of one white and one red ball is 2ⁿ/²ⁿCₙ.

RAW- F

1) From a pack of 52 cards, two are drawn one by one without replacement. Find the probability that both of them are kings.      1/221

2) From a pack of 52 cards, 4 are drawn one by one without replacement. Find the probability that all are aces (or Kings).      1/270725

3) Find the chance of drawing 2 white balls in succession from a bag containing 5 red and 7 white balls, the ball first drwan not being replaced.     7/22

4) A bag contentans 25 tickets, numbered from 1 to 25. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both the tickets will show even numbers.     11/50

5) From a deck of cards, 3 cards are drawn one by one without replacement. Find the probability that each time it is a card of spade.      11/850

6) Two cards are drawn without replacement from a pack of 52 cards . Find the probability that

a) both are kings.     1/221

b) the first is a king and the second is an ace.      4/663

7) A bag  contains 20 tickets, numbered from 1 to 20. Two tickets are drawn without replacement. What is the probability that the first ticket has an even number and the second an odd number.       5/19

8) An urn contains 3 white, 4 red and 5 black balls. Two balls are drawn one by one without replacement. What is the probability that atleast one ball is black ?     15/22

9) A bag contains 5 white, 7 red and 3 black balls. if three balls are drawn one by one without replacement. Find the probability that none is red.       8/65

10)  A card is drawn from a well suffled pack of 52 cards and then a second card is drawn. Find the probability that the first card is a heart and the second card is a diamond if the first card is not replaced.      13/204

11) An urn contains 10 black and five white balls. Two balls are drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black.        3/7

 

12) Three cards are drawn successively, without replacement from a pack of 52 well shuffled pack of 52 cards. What is the probability that the first two cards are kings and third card drawn is an ace ?      2/5525.

13) A box of oranges is inspected by examining 3 randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale otherwise it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and three are bad ones will be approved for sale.     44/91

14) A bag contains 4 white, 7 black and five Red balls. 3 balls are drawn one after the other without replacement. Find the probability that the balls drawn are white, black and the red respectively.        1/24

EXERCISE - G

1) If A and B are two events such that P(A)=0.5, P(B) = 0.6 and P(A U B)= 0.8, Find

A) (A/B).                                        1/2

B) P(B/A).                                      3/5

2) If A and B are two events such that P(A)=0.3, P(B) = 0.6 and P(B/A)= 0.5, Find

A) (A/B).                                        1/4

B) P(A U B).                                  0.75

3) If P(not A)= 0.7, P(B)= 0.7 and P(B/A)=0.5, then Find

A) (A/B).                                       3/14

B) P(A U B).                                  0.85

4) If A and B are two events associated with a random experiment such that P(A)=0.8, P(B) = 0.5 and P(B/A)= 0.4, Find

A) P(A∩B).                                  0.32

B) (A/B).                                     0.64

B) P(AUB).                                   0.98

5) A fair dice is rolled. Consider the following events:

A= {1,3,5}, B={2,3}, and C={2,3,4,5}. Find

A) P(A/B).                                     1/2

B) P(B/A).                                     1/3

C) P(A/C).                                     1/2

D) P(C/A).                                     2/3

E) P(A U B/C).                              3/4

F) P( A∩B/C).                               1/4

6) Three events A, B , C have Probability 2/5,1/3, and 1/2 respectively. Given that P(A∩C)= 1/5 and P(B∩C)= 1/4, find

A) P(C/B).                                    3/4

B) P(A'∩C').                                3/10

7) P(A)= 3/8, P(B)= 1/2 and P(A∩B), find

A) P(A'/B').                                  3/4

B) P(B'/A').                                  3/5

8) a dice rolled twice in the sum of the numbers appearing on them is observed to be 7. What is the conditional probability that the number 2 has appeared atleast once? 1/3

9) A black and a red die are rolled.

A) find the condition probability of obtaining a sum greater than 9, given that the black die resulted in a 5.       1/3

B) Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.          1/9

10) Two Integers are selected at random from integers 1 through 11. If the sum is even, find the probability that both the numbers are odd.      3/5

11) 10% of the bulbs produced in a factory a red colour and 2% are red and defective. If one bulb is picked at random, determine the probably of its being defective if it is red. 1/5

12) A couple has two children. Find the probability that both are boys. if it is known that 

A) one of the children is a boy.      1/3

B) the older child is a boy.               1/2

13) Consider a random experiment in which a coin is tossed and if the coin shows head it is tossed again but if it shows a tail then a die is tossed. If 8 possible outcomes are equally likely, find the probability that the die shows a number greater than 4 if it is known that the first throw of the coin results in a tail.       1/3

14) A coin is tossed twice and the four possible outcomes are assumed to be equally likely. If A is the event, ' both head and tail have appeared ' and B be the event, ' atmost one tail is observed', find

A) P(A).        1/2

B) P(B).         3/4

C) P(A/B).      2/3

D) P(B/A).        1

15) A bag contains 3 red and 4 black balls and another bag has 4 red and 2 black balls. One bag is selected at random and from selected bags ball is drawn. Let A be the event that the first bag is selected B be the event that the second bag selected and C be the event that the ball drawn is red. find

A) P(A).        1/2

B) P(B).        1/2

C) P(C/A).     3/7

D) P(C/B).     2/3

16) A coin is tossed, then a die is thrown. Find the probability of obtaining a '6' given that head come up.         1/6

17) A committee of 4 Students is selected at random from a group consisting of 8 boys and 4 girls. Given that there is at least one girl in the committee, Calculate the probability that there are exactly two girls in the committee.      168/425

18) Two coins are tossed. What is the probability of coming up two heads if it is known that at least one heads comes up.      1/3

19) An instructors has a test bank consisting of 300 easy True/false questions. 200 difficult True/false questions, 500 easy multiple choice questions (MCQ) and 400 difficult multiple choice questions. If a questions is selected at random from the test bank, what is the probability that it will be an easy question given it is a multiple choice question.      5/9

20) A die is thrown three times. Events A and B are defined as follows:

A: 4 on the third throw, 

B: 6 on the first and 5 on the second throw.

Find the probability of A given that B has already occured.     1/6

21) Three dice are thrown at the same time. Find the probability of getting three two's if it is known that the sum of the numbers on the dice was a 6.    1/10

22) In a hostel 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random.

A) find the probably that she reads neither Hindi nor English news papers.   1/5

B) If she reads Hindi newspaper, find the probability that she reads English newspaper. 1/3

C) if she reads English newspaper, find the probability that she reads Hindi newspaper. 1/2

23) An electronic assembly consists of two subsystems say A and B. From previous testing procedures the following probability are assumed to be known.

P(A fails)=0.2, P(B fails alone)= 0.15, P(A and B fail)= 0.15. find

A) P(A fails/B failed).     1/2

B) P(A fails alone).      0.05

24) Three distinguishable balls are distributed in three cells. Find the conditional probability that all three occupy the same same cell, given that at least two of them are in the same cell.        1/7

25) ** Consider the experiment of tossing a coin. If the coin shows head toss it again but if is shows tail then throw a die. find the conditional probability of the event ' the die shows a number greater than 4, given that 'there is at least one tail'.     2/9

26) Consider the experiment of throwing a die, if a multiple of 3 comes up throw the die again and if any other number comes toss a Coin. Find the conditional probability of the event 'the coin shows a tail', given that 'at least one die shows a 2'.     3/8

RAW- G

1) P(A)=7/13, P(B) = 9/13 and P(A∩ B)= 4/13, Find (A/B).      4/9

2) If A and B are two events such that P (A)= 0.6, P(B)= 0.3 and P(A∩ B)= 0.2, Find

a) P(A/B).     2/3

b) P(B/A).    1/3

3) If A and B are two events such that P(B)= 0.5 and P(A∩ B)= 0.32, Find P(A/B).  0.64

4) If P (A)= 0.4, P(B)= 0.8 and P(B/A)= 0.6, Find

a) P(A/B).                                       0.3

b) P(A UB).                                    0.96

5) If A and B are two events such that 

a) P (A)= 1/3, P(B)= 1/4 and P(AU B)= 5/12, Find

i) P(A/B).                                       2/3

ii) P(B/A).                                      1/2

b) P (A)= 6/11, P(B)= 5/11 and P(AU B)= 7/11, Find

i) P(A∩ B).                                   4/11

ii) P(A/B).                                     4/5

iii) P(B/A).                                     2/3

c) P (A)= 7/13, P(B)= 9/13 and P(A∩ B)= 4/13, find P(A'/B). 5/9

d) P (A)= 1/2, P(B)= 1/3 and P(A∩ B) = 1/4 find

i) P(A/B).                                  3/4

ii) P(B/A).                                   1/2

iii) P(A'/B).                                  1/4

iv) P(A'/B').                                  5/8

6) If A and B are two events such that 2P (A)= P(B)= 5/13 P(A/B)= 2/5 Find P(AUB). 11/26

7) P (A)= 6/11, P(B)= 5/11 and P(AU B)= 7/11, Find

a) P(A∩ B).                                    4/11

b) P(A/B).                                      4/5

c) P(B/A).                                     2/3

8) A coin is tossed three times. Find P(A/B) in each of the following:

a) A= heads on 3rd toss,

     B= Heads on the first toss at least two heads at most two tosses.     1/2

b) A: atleast two heads,

    B: at most two heads.            3/7

c) A: atmost two tails,

     B: at least one tail.          6/7

9) Two coins are tossed once. Find P(A/B) in each of the following

a) A= tail appears on one coin,

     B= One coin shows head.      1

b) A= no tail appears

     B= No head appears.          0

10) A die is thrown 3 times. Find

a) P(A/B).                                     1/6

b) P(B/A).                                     1/36

IF A= 4 appears on the third toss

    B= 6 and 5 appears respectively on first two tosses

11) Mother, father and son line up at random for a family picture. if A and B are two events given by 

A= son of one end,

B= father in the middle, find

a) P(A/B).                                        1 

b) P(B/A).                                     1/2

12) A dice is thrown twice and the sum of the numbers appearing is observed to be 6. What is the conditional probability that the number 4 has appeared at least once? 2/5

13) Two dice are thrown. Find the probability that the numbers appeared has the sum 8, if it is known that the second die always exhibit 4.                                      1/6

14) A pair of dice is thrown. Find the probability of getting 7 as the sum, if it is known that the second die always exhibits an odd number.   1/6

15) A pair of dice is thrown. Find the probability of getting 7 as the sum of it is known that the second die always exhibits a prime number.    1/6

16) A dice is rolled. if the outcome is an odd number, what is the probability that it is prime.     2/3

17) A pair of dice is thrown. Find the probability of getting the sum 8 or more, if appears on the first die.      1/2

18) Find the probability that the sum of the numbers showing on two dice is 8, given that at least one die does not show five.      3/25

19) Two numbers are selected at random from integers 1 through 9. If the sum is even, find the probability that both numbers are odd.     5/8

20) A dice is thrown and the sum of the numbers appearing is observed to be 8. What is the conditional probability that the number 5 has appeared atleast once?         2/5

21) Two dice are thrown and it is known that the first die shows a 6. Find the probability that the sum of the numbers showing on two dice is 7.                                             1/6

22) A pair of dice is thrown. Let E be the event that the sum is greater than or equals to 10 and F be the event "5 appears on the first die". Find P(E/F). If F is the event " 5 appears atleast one die", find P(E/F).     1/3, 3/11

23) The probability that a students selected at random from a class will pass in mathematics is 4/5, and the probability that/he/she passes in Mathematics and Computer Science is 1/2. What is the probability that he/she will pass in Computer Science if it is known that he/she has passed in Mathematics?             5/8

24) The probability that a certain will buy a shirt is 0.2, the probability that he will buy a trouser is 0.3, and the probability that he will buy a shirt given that he buys a trouser is 0.4. Find the probability that he will buy both a shirt and a trouser. Find also the probability that he will buy a trouser given that he buys a shirt.    0.12, 0.6

25) In a school there are 1000 students, out of which 430 are girls. It is known that out of 430, 10% the girls study in class XII. What is the probability that student chosen randomly studies in class XII given that the chosen students is a girl?     1/10

26) Ten cards numbered 1 through 10 are placed in a box, mixed up thoroughly and then one card is drawn randomly. If it is known that the number on the drawn card is more than 3, what is the probability that it is an even number?     4/7

27) Assume that each born child is equally likely to be a boy or girl. If a family has two children, what is the constitutional probability that both are girls? Given that

A) the youngest is a girl ?    1/2

B) at least one girl?                1/3

EXERCISE-H

1) A bag contains 4 white and 2 black balls. Another contains 3 white and 5 black balls. If one ball is drawn from each bag, Find the probability that

A) both are white.      1/4

B) both are black.       5/24

C) one is white and one is black.     13/24

2) A box contains 3 red and 5 blue balls. Two balls are drawn one by one at a time at random without replacement. Find the probability of getting 1 red and 1 blue ball. 15/28

3) Two cards are drawn from a well shuffled pack of 52 cards without replacement. What is the probability that one is a red queen and the other is a king of black colour.  2/663

4) 20) Two cards are drawn without replacement from a well shuffled pack of 52 cards. Find the probability that one is a spade and other is a queen of red colour.     1/51

5) A box contains 5 white and 3 black balls. Four balls are drawn one by one at a time at random without replacement. What is the probability that they are alternatively of different colours. 1/7

6) Cards are numbered 1 to 25. Two cards are drawn one after the other. Find the probability that the number on one card is multiple of 7 and on the other it is a multiple of 11.    1/50

7) A bag contains 4 red and 5 black balls. Another contains 3 red and 7 black balls. If one ball is drawn from 1st bag and two balls drawn from 2nd bag, find the probability that out of 3 balls drawn, two are black and one is red.            7/15

8) A bag contains 5 red and 3 black marbles. Three marbles are drawn one by one at a time at random without replacement. Find the probability that atleast one of the three marbles drawn be black, if the first marble is red.       25/56

9) Three groups of children 3 girls and 1 boy; 2 girls and 2 boys; 1 girl and 3 boys respectively. One child is selected at random from each group. Find the chance that the three selected comprise one girl and 2 boys.     13/32

10) The probabilities of A, B and C solving a problem are 1/3, 2/7 and 3/8 respectively. If all the three try to solve the problem simultaneously, find the probability that exactly one of them can solve it.      25/56

11) 3 critics review a book. Odds in favour of the book are 5:2,  4:3 and 3:4 respectively for 3 critics. Find the probability that the majority are in favour of the book.   209/343

12) A, B and C shot to hit a target. If A hits the target 4 times in trials; B hits it 3 times in 4 trials and C hits 2 times in 3 trials; what is the probability that the target is hit by atleast 2 persons.     5/6

13) A speaks truth in 60% of the cases and B in 90% of the cases. In what percentage of cases are they likely to 

a) contradict each other in stating the same fact?      42/100

b) agree in stating the same fact?      58%

14) The odds against a husband who is 45 years old, living till he is 70 are 7:5 and the odds against his wife who is now 36, living till she is 61 are 5:3. Find the probability that 

a) the couple will be alive 25 years hence,       5/32

b) exactly one of them will be alive 25 years hence.     23/48

c) none of them will be alive 25 years hence.      35/96

d) atleast one of them will be alive 25 years hence.      61/96

15) A box contains 3 white, 3 black and 2 red balls. Three balls are drawn one by one at a time at random without replacement. Find the probability that the third ball is red. 1/4

16) The probability of student A passing an examination is 3/7 and of student B passing is 5/7. Assuming the two events "A passes", "B passes", as independent, find the probability of:

a) Only A passing the examination.     6/49

b) Only one of them passing the examination.      26/49

17) There are three bag A, B and C. Bag A contains 4 white and 5 blue balls. Bag B contains 4 white and 3 blue balls. Bag C contains 2 white and 4 blue balls. If one ball is drawn from each bag, find the probability that out of these three balls drawn, two are white and one is a blue ball.      64/189

18) A certain team wins with probability 0.7, loses with probability 0.2 and ties with probability 0.1 the team plays three games. Find the probability that the team wins atleast two of the games, but not lose.         0.49

19) A clerk was asked to mail three report cards to three students. He addresses three envelopes but unfortunately paid no attention to which report cards be put in which envelope. What is the probability that exactly one of the students received his or her own card ?      1/2

20) Robin is taking up subjects mathematics, physics and chemistry. She estimates that her probabilities of receiving grade A in these courses are 0.2,0.3 and 0.9 respectively. If the grades can be regarded as independent events, find the probabilities that she receives 

a) All A's.     0.054

b) No A's.      0.056

c) Exactly two A's.       0.348

21) A doctor claims that 60% of the patients he examines are allergies to some type of weed. What is the probability that 

a) exactly 3 of his next 4 patients are allergic to some type of weeds?   216/625

b) none of his next 4 patients is allergic to weeds ?     16/625

22) If each element of a second order determinant is either zero or one, what is the probability that the value of the determinant is positive? (Assuming that the individual entries of the determinant are chosen independently, each value being assumed with probability 1/2).      3/16

23) An electric system has open-closed switches X, Y and Z as shown in figure. The switches operate independently of one another and the current will flow from A and B either if X is closed or if both Y and Z are closed. If P(X)= P(Y)= P(Z)= 1/2, find the probability that the circuit will work.       5/8

24) Two persons A and B throw a die alternatively till one of them gets a 'three' and wins the game. Find their respectively probabilities of winning, if A begins.     5/11

25) A and B throw alternatively a pair of dice. A wins if he throws 6 before B throws 7 and B wins if the throws 7 before A throws 6. Find their respective chance of winning, if A begins.       31/61

26) Three A, B and C throw a die in succession till one gets a 'six' and wins the game. Find their respective probabilities of winning, if A begins.      25/91

RAW- H

1) A bag contains 6 black and 3 white balls. Another bag contains 4 white and 5 black balls. If one ball is drawn from each bag, find the probability that these two balls are of the same colour.     14/27

2) A bag contains 3 red and 5 black balls. Another bag contains 6 red and 4 black balls. If one ball is drawn from each bag, find the probability that one is red and the other is black.    21/40

3) Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that 

a) both the balls are red.     16/81

b) first ball is black and second is red.     20/81

c) one of them is black and other is red.     40/81

4) Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability of exactly one ace.       32/221

5) A speak truth in 75% and B in 80% of the cases. In what percentage of cases are they likely to contradict each other in narrating the same incident.    35%

6) Richa and Riya appeared for an interview for two vacancies. The probability of Richa's selection is 1/3 and that of Riya's selection is 1/5. Find the probability that

a) both of them will be selected.    1/15

b) none of them will be selected.      8/15

c) atleast one of them will be selected.     7/15

d) only one of them will be selected.     2/5

7) A box contains 3 white, 4 red and 5 black balls. Two balls are drawn one by one at a time at random without replacement. Find the probability of getting one is white and other is black.     5/22

8) A bag contains 8 red and 6 green balls. Three balls are drawn one by one at a time at random without replacement. Find the probability that atleast two balls drawn are green. 5/13

9) Arun and Richard appeared for an interview for two vacancies. The probability of Arun's selection is 1/4 and that of Richard's rejection is 2/3. Find the probability that atleast one of them will be selected.      1/2

10) A and B toss a coin alternatively till one of them gets a head and wins the game. If A starts the game, find the probability that B will win the game.      1/3

11) Two cards are drawn from a well shuffled pack of 52 cards, one after another without replacement. Find the probability that one of these is red card and the other a black card.      26/51

12) Tickets are numbered from 1 to 10. Two tickets are drawn one after the other at random. Find the probability that the number on one card is a multiple of 5 and on the other a multiple of 4.              4/45

13) In a family, the husband tells a lie in 30% cases and the wife in 35% cases. Find the probability that both contradict each other on the same fact.     0.44

14) A husband and wife appear in an interview for two vacancies for the same post. The probability of husband's selection is 1/7 and that of wife's selection is 1/5. What is the probability that 

a) both of them will be selected?     1/35

b) only one of them will be selected?     2/7

c) none of them will be selected?       24/35

15) A bag contains 7 white, 5 black and 4 red balls. Four balls are drawn one by one at a time at random without replacement. Find the probability of getting atleast three balls are black.      23/364

16) A, B and C are independent witness of an event which is known to have occurred. A speaks the truth three times out of four, B four times out of five and C five times out of six. What is the probability that the occurrence will be reported truthfully by majority of three witness?      107/120

17) A bag contains 4 white and 2 black balls. Another contains 3 white and 5 black balls. If one ball is drawn from each bag, Find the probability that 

a) both are white.      1/4

b) both are black.     5/24

c) one is white and one is black.     13/24

18) A box contains 4 white, 7 black and 5 red balls. Four balls are drawn with replacement. Find the probability that atleast two are white.    67/256

19) Three cards are drawn with replacement from a well shuffled pack of 52 cards. Find the probability that the cards are a king, a queen and a jack.     6/2197

20) A bag contains 4 red and 5 black balls. Another bag contains 3 red and 7 black balls. If one ball is drawn from each bag, find the probability that

A) balls are different colour.    43/90

B) balls are of the same colour.      47/90

21) A can hit a target 3 times in 6 shots, B : 2 times in 6 shots and C: 4 times in 4 shots. They fix a volley. What is the probability that atleast 2 shots hit?      2/3

22) The probability of student A passing an examination is 2/9 and of student B passing is 5/9. Assuming the two events: ' A passes ', B passes ' as independent, find the probability of 

a) only A passing the examination.      8/81

b) only one of them passing the examination.       43/81

23) There are three bag A, B and C. Bag A contains 4 red and 3 black balls. Bag B contains 5 red and 4 black balls. Bag C contains 4 red and 4 black balls. If one ball is drawn from each bag, find the probability that out of these three balls drawn, two are red and one is a black ball.     17/42

24) X is taking up subjects -- Mathematics, Physics and Chemistry in the examination. His probabilities of getting grade A in these subjects are 0.2,0.3 and 0.5 respectively. Find the probability that he gets 

a) Grade A in subjects.    0,03

b) Grade A in no subject.     0.28

c) Grade A in two subjects.     0.22

25) A and B take turns in throwing two dice, the first to throw 9 being awarded the prize. Show that their chance of winning are in the ratio 9:8.

26) A, B and C in order toss coin. The one to throw a head wins. What are their respective chances of winning assuming that the game may continue indefinitely.    4/7, 2/7,1/7

27) Three persons A, B and C throw a die in succession till one gets a six and wins the game. Find their respective probabilities of winning.      36/91,30/91,25/91

28) A and B turns in throwing two dice, the first to throw 10 being awarded the prize, show that if A has the first throw, their chance of winning are in the ratio 12:11.

29) A bag contains 3 red and 5 black balls. Another contains 2 red and 3 black balls. If one ball is drawn from 1st bag and 2 balls from 2nd bag. find the probability that out of the 3 balls drawn one is red and 2 are black balls.     39/80      

30) A bag contains 8 marbles of which 3 are blue and 5 are red. One marble is drawn at random, its colour is noted and the marble is replaced in the bag. A marble is again drawn from the bag and its colour is noted. Find the probability that the marble will be

A) blue followed by red.   15/64

B) blue and red in any order.    15/32

C) of the same colour.     17/32

31) An urn contains 7 red and 4 blue balls. Two balls are drawn one by one at a time at random with replacement. Find the probability of getting 

A) 2 red balls.    49/121

B) 2 blue balls. 16/121

C) one red and one blue ball. 56/121

32) A card is drawn from a well shuffled deck of 52 cards. The outcome is noted, the card is replaced and the deck is reshuffled. Another card is then drawn from the deck. Find the probability that

A) both the cards are of same suit.    1/4

B) first card is an ace and the second card is red queen.   1/338

THE LAW OF PROBABILITY 

EXERCISE- I

1) A bag contains 4 red and 3 black balls. A second bag contains 2 red and 4 black balls . One bag is selected random. From the selected bag, one ball is drawn. Find the probability that the ball drawn is red.     19/42

2) Find the probability of a drawing a one-rupee coin from a purse with two complements one of which contains 3 fifty -paise coins and 2 one-rupee coins and other 2 fifty -paise coins and 3 one-rupee coins.      1/2

3) One bag contains 4 white and 5 black balls. Another bag contains 6 white and 7 black balls. A ball is transferred from bag from the second bag and then a ball is drawn from the second bag. Find the probability that the ball drawn is white.     29/63

4) There are two bags. The first bag contains 5 white and 3 black balls and the second bag contains 3 white and 5 black balls. Two balls are drawn at random from the first bag are put into the second bag without noticing their colours. Then two balls are drawn from the second bag. Find the probability that the balls are white and black.     673/1260

5) A bag contains 6 red and 5 blue balls and another bag contains 5 red and 8 blue balls . A ball is drawn from the second bag and without noticing its colour is put in the second bag. A ball is then drawn from the second bag. Find the probability that the ball drawn is blue in colour.           93/154

6) There are two bags, 1 of which contains 3 black and 4 white, balls, while the other contains 4 black and 3 white balls. A fair die is cast, if the face 1 or 3 turns up, a ball is taken from the first bag , and if any other face turns up a ball is chosen from the second bag. Find the probability of choosing a black ball.       11/21

7) A bag contains 4 black and 6 red balls and bag B contains 7 black and 3 red balls. A die is thrown. If 1 or 2 appears on it, then bag A is chosen, otherwise bag B. If two balls are drawn at random (without replacement) from the second bag, find the probability of one of them being red and another black.        22/45

8) Two-thirds of the students In a class are boys and the rest girls. It is known that the probability of a girl getting a first class is 0.25 and that of a boy getting a first class is 0.28. Find the probability that a student chosen at random will get first class marks in the subject.    0.27

9) In the bolt factory, machines A, B and C manufacture respectively 25%, 35% and 40% of the total bolt. If their output 5, 4 and 2 percent are respectively defective bolts. A bolt is drawn at random from the product. What is the probability that the bolt drawn is defective ?     0.0345

10) A person has undertaken a construction job. The probability are 0.65 that there will be strike, 0.80 that the construction job will be completed on time if there is no strike, and 0.32 that the construction job will be completed on time if there is a strike, Determine the probability that the construction job will be completed on time.      0.488

11) A box has 5 blue and 4 red balls. One ball is drawn at random and not replaced. Its colour is also noted. Then another ball is drawn at random. What is the probability of second ball being blue?      5/9

12) An urn contains m white and n black balls . A ball is drawn at random and is put back into the urn along with k balls of the same colour as that of the ball drawn . A ball is again drawn at random. Show that the probability of drawing a white ball now does not depend on k.

13) A bag contains (2n +1) coins. It is known that n of these coins have a head on both sides whereas the rest of the coins are fair. A coin is picked up at random from the bag and is tossed. If the probability that the toss results in a head is 31/42, determine the value of n.         10

14) Three bag contains a number of red and white balls as follows:

Bag I: 3 red balls; Bag II: 2 red balls and 1 white ball; Bag III: 3 white balls.

The probability that the bag i will be chosen and a ball is selected from it is i/6, i= 1,2,3. What is the probability that 

a) a red ball will be selected?       7/18

b) a white ball will be selected?      11/18