AREA. (COMPETITIVE EXAMS)

EXERCISE 1

1) P(a, f(a)) and Q(b, f(b)) are ends of an arc in the first quadront. The area bounded by the arc, coordinate through P and Q, and the x-axis is

A) ∫ f⁻¹(y) dy at (f(b), f(a))

B) ∫ f⁻¹(y) dy at (b, a)

C) ∫ f(x) dx at (b, a)

D) ∫ f(x) dx at (f(b), f(a)) 

2) The area bounded by the lines y=|x - 2|, |x| = 3 and y= 0 is 

A) 13unit² B) 5 unit² C) 9 unit² D) 7 unit²

3) The area bounded by the curve y= √(4- x²) and the line y= 0 is 

A) 4π.  B) 2π.  C) π.      D) π/2 

4) The area bounded by the curve y² = 4x and the double ordinate x= 2, is 

A) 4√2/3 unit²    B) 8√2/3 unit² C) 16√2/3 unit²       D) 30√2/3unit²

5) The area bounded by the curve x² = ky, k > 0 and the line y= 3 is 12 unit². Then k is

A) 3     B) 3√3     C) 3/4      D) none

6) The area bounded by the curve y= 2ˣ , the x-axis and the y-axis.

A) logₑ2 B) log ₑ4 C) log₄e D) log ₂e

7) The area of the portion enclosed by the curve √x +√y = √a and the axes of reference is

A) a²/6     B) a²     C) a²/2     D) a²/4

8) The area bounded by the curve x= cos cos⁻¹y and the lines | x|= 1 is

A) sin 1    B) cos 1    C) 2 sin 1     D) 2cos 1

9) The area of bounded by the curve y=√x, the line 2y +3= x and the x-axis in the first quadrant is

A) 9     B) 27/4      C) 36     D) 18 

10) the area of the region bounded by the pair of lines y=|x -1| and y= 3 - |x| is

A) 3 unit²    B) 4 unit²    C) 6unit²   D) 2 unit²

11) The ratio in which the area bounded by the curves y² = 4x and x²= 4y is divided by the line x= 1 is

A) 64: 49   B) 15: 34   C) 15:49   D) n

12) Let f(x) be a continuous function such that the area bounded by the curves y= f(x), the x-axis and the two ordinates x= 0 and x= a is a²/2 + a/2 sin a + π/2 cos a. Then f(π/2) is

A) 1/2     B) π²/8 +π/4       C) (π+1)/2      D) π/2

13) the area bounded by the curve f(x)= ceˣ (c> 0), the x-axis and the two ordinates x = p and x= q is proportional to

A) f(p).f(q)     B) |f(p) - f(q)|     C) f(p) + f(q)       D) √f(p). f(q)

14) The ordinate x= a divides the area bounded by the x-axis, the curve y= 1 + 8/x² and the ordinats x= 2 and x= 4 into equal parts. Then a is.

A) 3     B) 7/2     C) 2√2      D) 5/2

15) The area of the ellipse (x+1)²/4 + y²= 1 falling in the first quadrant is

A) (4π - 3√3)/6 B) (4π - 3√3)/12 C) √3/2. D) (π - √3)/3

ANSWERS

1) C 2) A 3) B 4) C 5) A 6) D 7) A 8) C 9) A 10) B 11) C 12) A 13) B 14) C 15) B

EXERCISE -2

1) The area enclosed by the curve y= x², the straight line y= x +2 and the x-axis is

A) 5/6 sq. unit B) 5/4sq. unit C) 5/2 sq. unit D) none

2) The area bounded by the curve y²+ 6x= 0 and y² + 4x= 4 is 

A) 4/√3 sq. unit B) 8/√3 sq. unit C) 4√3 sq. unit D) none

3) The area bounded by y= |x -1|, |x|= 2 and the x-axis is

A) 3 sq. unit B) 4 sq. unit C) 5 sq. unit D) none

4) The area bounded by the lines |x|+|y|= 1 is 

A) 15sq. unit B) 2sq. unit C) 3sq. unit D) 4sq. unit 

5) The area bounded by y=|x|, |x|= 1and the x-axis is

A) 1sq. unit B) 2sq. unit C) 3sq. unit D) 4 sq. unit 

6) The area bounded by y=|x -1| and y= 1 is

A) 1/2sq. unit B) 1 sq. unit C) 2sq. unit D) 4 sq. unit 

7) The area bounded by y= 1 - |x| and the x-axis is

A) 1/3sq. unit B) 1/2 sq. unit C) 1 sq. unit D) 2sq. unit 

8) The area bounded by y = |sinx|, the x-axis and the lines |x| =π/2 is

A) 1 sq. unit B) 1/2 sq. unit C) 2 sq. unit D) none

9) The area bounded by the lines y= |x -1| and y= 3 - |x| is

A) 1 sq. unit B) 2 sq. unit C) 3 sq. unit D) 4 sq. unit

10) The area included between y=√(8- x²) and y=|x| is equals to

A) π sq.unit B) 3π/2 sq unit C) 2π sq unit D) 3/2 (π +1) sq unit

11) The smaller area included between x²+ y²= a² and the line x+y = a is

A) a²/4 (π -2) sq. unit B) a²/4 (2 - π)sq. unit C) a²/4 (π +2) sq. unit D) none

12) The smaller area included between y=√(4 - x²), y= x √3 and the x-axis is 

A) π/3 sq. unit B) 2π/3 sq. unit C) 4π/3 sq. unit D) none

13) The area included between x² + y²= 2ax and y²= ax is

A) a²/8 (3π - 8) sq. unit B) a²/6 (3π +8)sq. unit C) a²/6 (3π -8) sq. unit D) none

14) The area included between y²= 4x and x²= 4y is

A) 20/3 sq. unit B) 32/3 sq. unit C) 16/3 sq. unit D) none

15) The area bounded by y= x², y= 0, x= 2, x= 4 is.

A) 6 sq. unit B) 12 sq. unit C) 53/3 sq. unit D) 56 sq unit

16) T area bounded by a²= 4x, x= 0, and y= 2 is

A) 3/2 sq. unit B) 2/3 sq. unit C) 3 sq. unit D) none 

17) The area bounded by y= x² and y= x³ is

A) 1/12 sq. unit B) 1/6 sq. unit C) 1/24 sq. unit D) none

18) The area bounded by the parabola the y²= 4a(x +a) and y²= -4(x - a), a > 0, is

 A) 16a²/3 sq. unit B) 8a²/3 sq. unit C) 4a²/3 sq. unit D) none

19) The area of the region bounded by y= |[x - 1]|, the x-axis and the lines x =|2|

A) 6 sq. unit B) 8 sq. unit C) 4 sq. unit D) none

Answers

1) A 2) B 3) C 4) B 5) A 6) B 7) C 8) C 9) D 10) C 11) A 12) B 13) C 14) C 15) D 16) B 17) A 18) A 19) D

EXPONENTIAL (VIII)

EXERCISE -1

1) 64¹⁾².                                            8

2) 32¹⁾⁵.                                            2

3) 125¹⁾³.                                          5

4) 9³⁾².                                             27

5) 32²⁾⁵.                                              4

6) 16³⁾⁴.                                               8

7) 125 ⁻¹⁾³.                                     1/5

8) 2²⁾³ c2¹⁾⁵.                                 2¹³⁾¹⁵

9) (1/3³)⁷.                                   1/3²¹

10) 11¹⁾²/ 11¹⁾⁴.                          11¹⁾⁴

11) 7¹⁾². 8¹⁾².                            (56)¹⁾²

12) 4³⁾².                                              8

13) (8¹⁾⁵)⁴.                                      8⁴⁾⁵

14) (1/3⁻⁵)⁴.                                    3⁴⁾⁵

15) 7⁴⁾⁵/7¹⁾³.                                7⁻²⁾¹⁵

16) 13¹⁾¹⁵÷ 13¹⁾³⁰.                    13¹⁾³⁰   

17) 2²⁾³.2¹⁾³.                                    2

18) 13¹⁾⁵. 17¹⁾⁵.                        221¹⁾¹⁵

19) 16³⁾².                                          64

20) 8¹⁾³/8¹⁾⁶.                                  8¹⁾⁶

21) 3²⁾³. 8²⁾³.                                24²⁾³

22) (2³)².                                           64

23) 625¹⁾⁴.                                          5

24) (64)⁻¹⁾³.                                    1/4

25) ³√(64)⁻².     

A) 1/2 B) 1/8 C) 1/4

26) (√x³)²⁾³

A) x B) 1/x C) x²

27) ⁴√{³√(5²)}.

A) 5¹⁾⁶ B) 5⁻⁶ C) 5¹⁾⁶

28) If x and y are two natural numbers such that x² = 32 then ᵧx² is

A) 5² B) 5³ C) 5⁴

29) If (2)3)⁶x (9/4)⁵ = (3/2)ᵐ⁺²

A) 9 B) -2 C) 2

           

            

EXERCISE- 2

1) (1³ + 2³ + 3³)¹⁾²

A) 2 B) 3 C) 6 D) N

2) (5¹⁾³ x 25¹⁾³ x 125²⁾³)/625¹⁾²

A) 5 B) 25 C) 625 D) 1

3) [{(81)¹⁾²}¹⁾⁴]².

A) 3 B) 3¹⁾² C) 3¹⁾⁶ D) none

4) (2/3)⁴ (8/3)⁻¹² (32/3)⁶

A) 4/3 B) 4/9 C) 9/4 D) 9/16

5) (81)³⁾⁴ - (1/32)⁻²⁾⁵ + 8²⁾³ x (1/2)⁻¹ x 3⁰ - (1/81)⁻¹⁾² 

A) 22 B) 24 C) 26 D) 28

6) (64/125)⁻²⁾³ + 4⁰ x 9⁵⁾² x 3⁻⁴ - √25/ ³√64 x (1/3)⁻¹ 

A) 1/2 B) 1/6 C) 13/16 D) none

7) (1/4)⁻² - 3x8²⁾³ x 5⁰ + (9/25)⁻¹⁾²

A) 1/3 B) 16/3 C) 3/16 D) none

8) √(1/4) + (0.01)⁻¹⁾²- (27)²⁾³x3⁰ 

A) 2/3 B) 3/2 C) 1/2 D) 1/3

9) 4/(216)⁻²⁾³ - 1/(256)⁻³⁾⁴

A) 4 B) 64 C) 80 D) No

10) {(2⁻¹ x 3²)/(2²x 3⁻⁴)}⁷⁾² x {(2⁻² x 3³)/(2³x 3⁻⁵)}⁻⁵⁾² 

A) 1 B) 2 C) 12 D) 21

11) [(64)²⁾³ x 2⁻² ÷ 7⁰]⁻¹⁾². 

A) 1 B) 2 C) 1/2 D) none

12) (81/16)⁻³⁾⁴ x[(25/9)⁻³⁾²÷(5/2)⁻³]

A) 1 B) 0 C) 1/2 D) none

13) (32)⁻⁴⁾⁵ / (27)⁻²⁾³

A) 9/16 B) 16/9 C) 3/4 D) 4/3

14) (8)²⁾³. (16)³⁾⁴ . (32)⁻⁴⁾⁵. 

A) 2 B) 4 C) 8 D) none

15) {(125)⁻³. (64)⁻³⁾²} ⁻¹⁾⁹. 

A) 20 B) 40 C) 60 D) none 

16) (2⁰ + 7⁰)/5⁰

A) 2 B) 7 C) 5 D)1 E) none

17) (2³⁰ + 2²⁹)/(2³¹ - 2³⁰)

A) 1 B) 2 C) 2/3 D) 3/2

18) √(3⁻²)

A) 1/9 B) 9 C) - 3 D) 1/3

19) (12)¹⁾³. (36)¹⁾⁴ /(96)⁻¹⁾⁶. 

PROBABILITY(Basic - 2)

RECAPITULATION

Let us recall important terms and concepts which we have studied in earlier

Random experiment :

If an experiment, when repeated under identical conditions, do not produce the same outcome every time but the outcome in a trial is one of the several possible outcomes, than such an experiment is called a random experiment or a probabilistic experiment.

ELEMENTARY EVENT 

If a random experiment is performed, then each of its outcomes is known as a element as an elementary event.

SAMPLE SPACE 

The set of all possible outcomes of a random experiment called the sample space associated with it.

EVENT 

A subset of the sample space associated with a random experiment is called an event.

OCCURRENCE OF AN EVENT  

An event associated to a random experiment is said to occur if any one of the alimentary in events belong to it is an outcome.

      corresponding to every event A, associated to a random experiment, we define an event "not A denoted by A' "which is said to occur when and only when A does not occur.

  CERTAIN (OR SURE) EVENT 

An event associated with a random experiment is called a certain event if it always occur whenever the experiment.is performed.

Impossible event 

An event associated with a random experiment is called an impossible event if it occurs wherever the experiment is performed.

 Compound event 

An event associated with a random experiment is a compound event, if it is the disjoint union of two or more elementary events.

Mutually exclusive events :

Two or more events associated with a random experiment are said to be mutually exclusive or impossible events if the occurrence of any one of them prevents the occurrence of all others, i.e., if no two or more of them can occur simultaneously in the same trial.

Exhaustive event 

Two or more events associated with a random experiment are exhaustive if their union is the sample space.

Favourable element elementary events:

 Let S be the sample space associated with a random experiment and A be an event associated with the experiment. Then, elementary events belonging to A are known as favourable alimentary events to the event A.

Thus, an elementary event E is a favourable to an event A, if the occurrence of E ensures the happening or occurrence of event A.

Events associated to a random experiment are generally described verbally and it is very important to have the ability of conversion of verbal description to equivalent set theoretic notations. 

Following the table provides verbal description of some events and their equivalent set theoretic notations for ready reference.

Verbal Equivalent       Equivalent

description                        set

of the event                   theoretic

                                          notation 

 --------------                         -------------

• Not A.                                   A'

• A or B(atleast one

of A or B)                           A U B

• A and B                           A ∩ B

• A but not B                     A ∩ B'

• B but not A                     A' ∩ B 

• neither A nor B              A' ∩ B'

• at least one of 

    A, B or C                    A U B U C

• Exactly one 

  of A and B          (A ∩ B')U (A' ∩ B)

• all three of A,B & C       A ∩ B∩ C

• exactly two

of A, B and C.           (A ∩ B ∩ C')U(A ∩ B' ∩ C)U (A' ∩ B ∩ C)

• Exactly one

of A, B and C.                   (A ∩ B' ∩ C')U(A' ∩ B ∩ C')U (A' ∩ B' ∩ C)

 PROBABILITY OF AN EVENT:

DEFINITION: 

if there are elementary events associated with the random experiment and m of them are favourable to an event A, then the probability happening or occurrence of A is denoted by P(A) and is defined at the ratio m/n.

Thus, P(A) = m/n

If P(A)=1, then A is is called the certain event and A is called impossible if P(A)= 0.

Also, P(A) + P(A')= 1

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

* If A and B are two events associated with a random experiment, then

1) P(AU B)= P(A)+ P(B) - P(A ∩ B)

If A and B are mutually exclusive events, then

2) P(AU B)= P(A)+ P(B) 

If A, B, C are three events associated with random experiment, then

3) P(A U BU C)= P(A)+ P(B) + P(C) - P(A ∩ B) - P(B ∩ C)- P(A ∩ C) + P(A ∩ B ∩ C).

If A, B, C are mutually exclusive events, then

P(A U BU C)= P(A)+ P(B) + P(C)  

For any two events A and B, we have

I) Probability of occurrence of A only = P(A ∩ B')= P(A) - P(A ∩ B).

II) Probability of occurrence of B only = P(A' ∩ B)= P(B) - P(A ∩ B).

III) Probability of occurrence of exactly one of A and B = P(A ∩ B')= P(A' ∩ B)= P(A)+ P(B)- 2P (A ∩ B)= P(A U B) - P(A ∩ B).

Note:

* P(A)= P(A ∩ B)+  P(A ∩ B').

* P(B)= P(A ∩ B)+ P(A' ∩ B).

* P(A U B)= P (A ∩ B)+ P(A ∩ B') + P(A' ∩ B)

Condition Probability:

Let A and B be two events associated with a random experiment. Then, the probability of occurrence of event A under the condition that B has already occured and P(B)≠ 0, is called the conditional probability and it is denoted by P (A/B). Thus, we have

          P (A/B)= Probability of occurrence of A given that B has already occured.

Similarly, P (B/A) when P (A)≠ 0 is defined as the probability of occurrence of event B when A has already occured.

In fact, the meaning of symbols P (A/B) and P (B/A) depend on the nature of the events A and B and also on the nature of the random experiment. These two symbols have the following meaning also.

P (A/B)= Probability of occurrence of A when B occurs      OR

Probability of occurrence of A when B is taken as the sample space 

      OR

Probability of occurrence of A with respect to B.

And, 

P (B/A)= Probability of occurrence of B when A occurs.        OR

Probability of occurrence of B when A is taken as the sample space

          OR

Probability of occurrence of B with respect to A

1) P(A/B)= n(A ∩ B)/n(B) 

2) P(B/A)= P(A ∩ B)/P(A)

EXERCISE-1

1) A bag contains 7 red and 5 white balls. 4 balls are drawn at random. What is the probability that

A) all of them are red.                7/99

B) two of them would be red and two white.                                 14/33

2) 3 cards are drawn at random from a pack of 52 cards. Find the probability of getting

A) 2 aces.                              72/5525

B) 2 spades.                          117/850

C) 1 spade, 1 club, 1 diamond.     169/1700

D) 2 face cards.                  132/1105

E) Atleast 1 king.              1201/5525

3) Four cards are drawn at random from a full packet. Find the probability that two of them will be spades.                       4446/20825

4) An urn contains 9 balls, 2 of which are white, 3 blue and 4 black. 3 balls are drawn at random from the urn. What is the probability that

A) three balls will be of different colours.                                         2/7

B) 2 balls will be of the same colour and the third a different colours?     55/84

C) three balls will be of the same colour?                                        5/84

5) Two balls are drawn at random from a bag containing 6 white and 4 black balls. Find the chance that one is white and the other is black.    8/15

6) A box contains 10 balls of which 4 are black and 6 are red. Four balls are drawn at random. Find the probability of having exactly two red balls.                                    3/7

7) An urn contains 5 black, 6 red and 4 white balls. Five balls are drawn at random from the urn. Find the probability that exactly 2 of the drawn balls are black.      400/1001

8) One card is drawn from a full pack of 52 cards. Find the probability that the card drawn is

A) either a spade or a diamond. 1/2

B) either a spade or a king.       4/13

9) In a pack of 10 watches, 3 are known to be defective. If 2 watches are selected random from the pack, what is the probability that atleast one is defective.                      8/15

10) In a pack of 25 articles, 5 are defective. If 4 articles are selected random from the pack, what is the probability that exactly 2 of the drawn article are defective.  38/253

11) A number is chosen at random from the first 50 positive integers. Find the probability that the chosen number is divisible by 4 or 5.      2/5

12) The probability that a contract will get a plumbing contract is 2/3 and the probability that he will not get an electric contact is 5/9. If the probability of getting atleast one contact is 4/5. What is the probability that he will get both the contact?                                   14/45

13) A class consists of 30 boys and 20 girls of which half the boy and half the girls have blue eyes. Find the probability that a student chosen at random is a boy or has blue eyes.                                   4/5

14) From 200 tickets marked with the first 200 natural numbers, one is drawn at random; find the probability that it is multiple of 3 or 7.                         17/40

15) In a family there are 5 children. Find the probability that

A) all of them will have different birthdays.

B) two of them will have the same birthdays.

C) atleast two of them will have the same birthday.     

16) In a family there are 4 children. Find the probability that

A) all of them will have different birthdays.

B) two of them will have the same birthdays.  

17) It is known that a family has 3 children and atleast 1 of these 3 children is a boy. Find the probability that 2 of the 3 children are boys.                                      3/7

18) Two letters are drawn at random from the word HOME. Find the probability that,

A) both the letters are vowels.  1/6

B) atleast one is a vowel.        5/6

C) one of the letters chosen should be 'M'.                                     1/2

19) If the letters of the word RAMESH be arranged at random. Find the probability that there are exactly 3 letters between A and E ?

20) 4 boys and 2 girls occupy seats in a row at random. What is the probability that the girls occupy seats side by side?                     1/3

21) A 5 figure number is formed using the digits 0, 1, 2, 3,4 without repetition. Find the probability of it being divisible by 4.                   5/16

22) If the letters of the word TOWEL are arranged at random with no repetition, what is the probability that there are exactly two letters between O and E ?                     1/5

23) 5 men in a company of 20 are graduates. If 3 men are picked out of the 20 at random, what is the probability that they are all graduates? What is the probability of atleast one graduate?     1/114, 137/228

24) 5 students A, B, C, D, E occupy their seats at random in a bench. Find the probability that the students A and B are not consecutive.                               3/5

25) Five books on English and 3 books on mathematics are placed at random on a bookshelf. Find the probability that the books on mathematics are placed side by side.                 3/28

26) A executive committee of 6 is formed 4 ladies and 7 gentlemen. Find the probability that the committee will consist of

A) exactly 2 lady members.       5/11

B) atleast two lady members. 53/66

27) If the letters of the word PROBABILITY are arranged in random order, what is the probability that the word PROBABILITY gets formed again? (2!2!)/11!

28) Five commerce and four science students are arranged at random in a row. Find the probability so that the commerce and science students are placed alternatively.                         1/126

29) A five figure number is formed using the digits 5,6,7,8,0 without repetition. Find the probability of it being divisible by

A) 4.                                              5/16

B) 5.                                              7/16

30) There are 10 persons who are to be seated around a circular table. Find the probability that two particular person will always sit together.                                    2/9

31) Two unbiased dice are rolled together. Find the odds in favour of getting 2 digits, the sum of which. 1:5

32) The odds in favour of an event are 4: 3. The odds against another independent event are 2:3. What is the probability that atleast one of the events will occur?        29/35

33) Three balls are drawn at random from a bag containing 8 black and 10 white balls. What is the probability that they are all white? Also, find the odds in favour of the event and against the event.     5/34, 5:29, 29:5

EXERCISE- 2

1) A bag contains 10 white and 15 black balls. Two balls are drawn in succession without replacement. What is the probability that first is white and second is black.          1/4

2) A bag contains 5 white, 7 red and 8 black balls. If four balls are drawn in succession without replacement. What is the probability of getting all white balls.                           1/969

3) A bag contains 5 white and 8 black balls. Two successive drawing of three balls at a time without replacement. What is the probability that first draw gives 3 white and second 3 black balls.  7/429

4) A bag contains 2 white, 3 red and 4 black balls. Two balls are drawn in succession without replacement. What is the probability that atleast one ball is red.                           7/12

5) A bag contains 7 white and 5 red balls. Two balls are drawn in succession without replacement. What is the probability that both the balls are white.                         7/22

6) A bag contains 3 white, 4 red and 5 black balls. Two balls are drawn in succession without replacement. What is the probability that atleast one ball is black.                    15/22

7) A bag contains 5 white, 7 red and 3 black balls. Three balls are drawn in succession without replacement. What is the probability that none is red.                                              8/65

8) A bag contains 5 white and 10 black balls. Two balls are drawn in succession without replacement. What is the probability that both drawn balls are black ?              3/7

9) A bag contains 4 white, 5 red and 7 black balls. Three balls are drawn in succession without replacement. What is the probability that drawn ball are white, black and red respectively.                               1/24

10) Find the probability of drawing a diamond card in each of the two consecutive draws from a well shuffled pack of cards, if the card drawn is not replaced after the first draw.                                          1/17

11) From a pack of 52 cards, two cards are drawn one by one without replacement. Find the probability that both of them are kings.  1/221

12) From a pack of 52 cards, 4 cards are drawn one by one without replacement. Find the probability that all are aces(or, kings).  1/270725

13) From a pack of 52 cards, 3 cards are drawn one by one without replacement. Find the probability that each time it is a card of spade.  11/850

14) From a pack of 52 cards, two cards are drawn one by one without replacement. Find the probability that

A) both are kings.                     1/221

B) the first is a king and the second is an ace.                    4/663

C) the first is a heart and second is red. 25/204

15) From a pack of 52 cards, two are drawn one by one without replacement. Find the probability that the first card is a heart and the second is a diomond.           13/204

16) From a pack of 52 cards, three cards are drawn one by one without replacement. Find the probability that first two cards are kings and third card is an ace.              2/5525

17) A bag contains 19 tickets, numbered 1 to 19. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers.                 4/19

18) A bag contains 20 tickets, numbered 1 to 20. Two tickets are drawn without replacement. Find the probability that the first ticket has an even number and the second an odd number.            5/19

19) To test the quality 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

20) A box of oranges is inspected by examining three 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 3 are bad ones will be approved for sale.  44/91

EXERCISE-3

BASED ON:

1) P(A/B)= n(A ∩ B)/n(B)

2) P(B/A)= n(A ∩ B)/n(A)

3) P(A U B/C)= n(A UB ∩C)/n(C)

4) P(A ∩ B/C)= n(A ∩ B∩C)/n(C) 

1) 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

2) A coin is tossed 3 times. Find P(E/F) in each of the following:

I) E= Head on the third toss, F= Heads on first two tosses.         1/2

II) E= At least two heads, F=  at most two heads.                          3/7

III) E= at most two tails, F= at least one tail.                                       6/7

3) Two coins are tossed once. Find P(E/F) in each of the following:

A) E= tail appears on one coin, F= one coin shows head.                   1

B) E= no tail appears, F= No head appears.                                           0

4) 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

5) A couple has two children. Find the probability that

A) both the children are boys, if it is known that older child is a boy.    

B) both the children are girls, if it is known that the older child is a girl.

C) both the children are boys, if it is known that at least one of the children is a boy.           1/2, 1/2, 1/3

6) A pair of dice is thrown. If the numbers appearing on them are different, find the probability 

A) the sum of the number is 6.  2/15

B) the sum of the number is 4 or less.        1/15

C) the sum of the number is 4.  2/15

7) 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 atleast once.            2/5

8) A dice is thrown three times. Events A and B are defined as below:

A: Getting 4 on third due, B= Getting 6 on the first and 5 on the second throw.

 Find the probability of A given that B has already occurred.      1/6

9) 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.     1/4

B) A  sum 8, given that the red die resulted in a number less than 4.   1/3

10) Consider the experiment of throwing a dice, 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 three'.                  0

11) In a school, there at 1000 students, out of which 430 are girls. It is known that out of 430, 10% of the girls study in class XII. What is the probability that a student choosen randomly studies in class XII given that the student is a girl ?    1/10

12) 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 different multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be any easy questions given that it is a multiple choice question ?        5/9

13) 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 

A) P(A/B).                                    2/3

B) P(B/A).                                    1/3

14) If P(A)= 6/11, P(B)= 5)11 and P(A UB)= 7/11, Find

A)  P(A ∩ B).                                4/11

B) P(A/B).                                      4/5

C) P(B/A).                                    2/3

15) Evaluate  P(AUB), if P(A)= 2 P(B)= 5/13 and P(A/B)= 2/5.          11/26

16) Two integers are selected at random from integers 1 to 11. If the sum is even. Find the probability that both the numbers are odd.   3/5

17) A dice is thrown three times. If the first throw is a four, find the chance of getting 15 as the sum.    1/18

18) 10 cards numbered 1 to 10 are placed in a box, mixed 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

19) 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

20)  Given that two numbers are appearing on throwing two dyes are different. find the probability of event 'the sum of the numbers in the dice is 4'.                         1/15

21) A dice is tossed 3 times, find the probability that 4 appears on the third if it is given that 6 and 5 appear respectively on first two tosses.            1/6

22) find P(A/B), if P(B)= 0.5 and  P(A ∩ B)= 0.32.                         16/25

23) If P(A)= 0.4, P(B)= 0.3 and P(B/A) = 0.5, find

A) P(A ∩ B).                                  0.2

B) P(A/B).                                     2/3 

24) If A and B are two events such that P(A)= 1/3, P(B)= 1/5 And P(A U B)= 11/30, Find

A) P(A/B).                                      5/6

B) P(B/A).                                      1/2

 

25) A couple has two children. Find the probability that both the children are

A) males if it is known that at least one of the children is male.     1/3

B) females, if it is known that the elder child is a female.               1/2

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- 4

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

7) P(A)=7/13,  P(B) = 9/13 and P(A∩ B)= 4/13, Find (A/B).          4/9

8) 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

9) 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

10) 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

11) If A and B are two events such that 

I) P (A)= 1/3, P(B)= 1/4 and P(AU B)= 5/12, Find

A) P(A/B).                                      2/3

B) P(B/A).                                     1/2

II) P (A)= 6/11, P(B)= 5/11 and P(AU B)= 7/11, Find

A) P(A∩ B).                                  4/11

A) P(A/B).                                    4/5

B) P(B/A).                                    2/3

III) P (A)= 7/13, P(B)= 9/13 and P(A∩ B)= 4/13, find P(A'/B).    5/9

IV) P (A)= 1/2, P(B)= 1/3 and P(A∩ B) = 1/4 find

A) P(A/B).                                 3/4

B) P(B/A).                                  1/2

C) P(A'/B).                                 1/4

D) P(A'/B').                                 5/8

12) If A and B are two events such that  2P (A)= P(B)= 5/13 P(A/B)= 2/5 Find P(AU B).                    11/26

13) 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

14) 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

15) 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

16) 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

17) 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

18) 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

19) 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

20) 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

21) 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

22) A coin is tossed, then a die is thrown. Find the probability of obtaining a '6' given that head come up.                                                1/6

23)  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

24) 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

25) 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

26) 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

27) 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

28) 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

29) 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

30) 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

31) 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

32) 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

33) 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

34)  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

35) 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

36) 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

37) 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

38) 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

39) 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

40) 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

41) A dice is rolled. if the outcome is an odd number, what is the probability that it is prime.    2/3

42) A pair of dice is thrown. Find the probability of getting the sum 8 or more, if appears on the first die.  1/2

43) 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

44) 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

45) 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

46) 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

47) 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

48) 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

49) 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

50) 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

51) 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

52) 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-5

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 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

3) 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

4) 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

5) A bag contains 4 red and 5 black balls. Another 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

6) 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      

7) 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

8) 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

9) 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

10) 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

11) 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

12) A box contains 3 white and 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

13) A box contains 8 red and 10 black balls. Two balls are drawn one by one at a time at random with replacement. 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

14) 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

15) 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

16) 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

17) 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

18) 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

19) 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

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

21) 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

22) 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

23) Two cards are drawn successively without replacement from a well shuffled pack of 52 cards. Find the probability of exactly one ace.                32/221

24) 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

25) 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 multiple of 5 and on the other it is a multiple of 4.        4/45

26) A card is drawn from a well shuffled deck of 52 cards. The outcome is noted, the card is replaced and the deck is shuffled. 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 a red queen. 1/338

27) 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

28) Ajay and Bijay appeared for an interview for two vacancies. The probability of Ajay's selection is 1/4 and that of Bijay's rejection is 2/3. Find the probability that atleast one of them will be selected.      1/2

29) 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

30) Simran and shakshi appear in an interview for two vacancies for the same post. The probability of Simran 's selection is 1/7 and that of shakshi'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

31) 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

32) A and B tame 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

33) A, B and C in order toss a coin. The one to throw a head wins. What are their respective chances of winning assuming that the game may continue indrfinitely.    4/7, 2/7, 1/7

34) Three A, B, C throw a die in succession till one gets a 'six' and wins the game. Find their respective chances of winning.     36/91, 30/91, 25/91

35) A and B take 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.

36) Three groups of children contains 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 probability that the three selected comprise one girl and 2 boys.                         13/32

37) A, B, 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

38) 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%

B) agree in stating the same fact?  58%

39) The probability of students A passing an examination is 2/9 and of students 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

40) X is taking up subject - 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 all subjects. 0.03

B) Grade A in no subjects. 0.28

C) Grade A in two subjects. 0.22

41) Out of 100 students, two sections of 40 and 60 are formed. If you and your friend are among 100 students, what is the probability that:

A) you both enter the same section? 17/33

B) you both enter the different sections? 16/33

42) The probabilities of A, B, C solving a problem are 1/3, 2/7, 3/8 respectively. If all the three try to solve the problem simultaneously, find the probability that

A) exactly one of them solve it. 25/56

B) atleast one of them solve it

C) atleast two of them solve it

D) atmost one of them solve it

E) no one can solve it

F) exactly two of them solve it

G) problem will be solved.

43) A, B, C shot to hit a target. If A hits the target 4 times in 5 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

44) 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

45) 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

46) A clerk was asked to mail three report cards to three students. He addresses three envelopes but unfortunately paid no attention to which report card be put in which envelope. What is the probability that exactly one of the students received his or her own card? 1/2

47) Mahek is taking up subject mathematics, Accounts, English. She estimates that her probability of receiving grade A in these subjects are 0.2, 0.3, 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

48) A doctor claims that 60% of the patients he examines are allergic to some type of weed. What is the probability that

A) exactly 3 of his next 4 patients are allergic to weeds. 3/5

B) none of his next 4 patients is allergic to weeds. 16/624

49) 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. A:5/11, B: 6/11

50) 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 starts. A: 30/61, B: 31/61

Continue......

Miscellaneous -1 

1) A 4 digit number is formed using the digit 1, 2, 3, 5 with no repetitions. Write the probability that the number is divisible by 5. 1/4

2) When three dies are thrown, write the probability of getting 4 or 5 on each of the dice simultaneously. 1/27

3) Three digit numbers are formed with the digit 0, 2, 4, 6, and 8. Write the probability of forming a three digit number with the same digits. 1/25

4) A ordinary cube has four plane faces, one face marked 2 and 2 and another face marked 3, find the probability of getting a total of 7 in 5 throws.                                  5/6⁴

5) Three numbers are chosen from 1 to 20. Find the probability that they are consecutive.       18/20C3

6) 6 boys and 6 girls sit in a row at random. Find the probability that all the girls sit together.               1/132

7) If A and B are two independent events such that P(A)= 0.3 and P(A U B')= 0.8. Find P(B).              16/81

8) An unbiased dice with face marked 1, 2, 3, 4, 5, 6 is rolled four times. Out of 4 face values obtained, find the probability that the minimum face value is not less than 2 and the maximum face value is not greater than 5.               16/81

9) if A and B are two events write the expression for the probability of occurrence of exactly one of two events.          P(A)+ P(B)- 2P(A ∩B).       

10) Write the probability that a number selected at random from the set of first 100 natural numbers is a cube.                                1/25

11) In a competition A, B and C participating. The probability that A wins is twice that of B, the probability that B wins is twice that of C. Find the probability that A losses.                                   3/7

12) If A, B, C are mutually exclusive and exhaustive events associated to a random experiment, then write the value of P(A)+ P(B)+ P(C).      1

13) If two events A and B are such that P(A')= 0.3, P(B)= 0.4 and P(A ∩B') =0.5, find P(B/A' ∩B').         1/4

14) if A and B are ABC are two events, then write P(A ∩B') in terms of P(A) and P(B).

15) If P(A)= 0.3, P(B)= 0.6, P(B/A)= 0.5, find P(A UB).                       0.75

16) If A B and C if A and B are independent events such that P(A)= P(B)= P(C)= p, then find the probability of occurrence of atleast two of A, B and C.               2p² - 3p³

17) If A and B are independent events then write expression for P(exactly one of A, B occurs).        P(A) P(B') + P(B) P(A')

18) If A and B are independent events such that P(A)= p, P(B)= 2p and P(Exactly one of A and B occurs)= 5/9, find the value of p.    1/3, 5/12

Miscellaneous - 2

1) If one ball is drawn at random from each of the three boxes containing 3 white and 1 black, 2 white and 2 black, one white and three black balls, then the probability that 2 white and one black balls will be drawn is.    13/32

2)  A and B draw two cars each, one after another, from a pack of a well shuffled pack of 52 cards. The probability that all the four cards drawn are of the same colour.    44/(85x49)

3) A and B are two events such that P(A)= 0.25 and P(B)= 0.50. The probability of both happening together is 0.14. The probability of both A and B not happening is.         0.39

4) The probability of a student getting I, II ,III division in an examination are 1/10, 3/5, and 1/4 respectively. The probability that the student fails in the examination is..      27/100

5)  India play two matches each with West Indies and Australia. In any match the probabilities of India getting 0.1 and 2 points are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting atleast 7 points is.  0.0875

6) Three faces of an ordinary dice are yellow, two faces are red and one face is blue. The dice is rolled 3 times. The probability that yellow, red and blue face appear in the first ,second and third throws respectively, is.  1/36

7) The probability that a leap year will have 53 Fridays or 53 Saturdays is.       3/7

8) A person writes four letters and addresses 4 envelopes. If a the letters are placed in the envelopes at random, then the probability that all letters are not placed in the right envelopes is.  15/24

9) A speaks truth in 75% cases and B speaks 80% cases. probability that they contradict each other in a statement is.   7/20

10) Three integers are chosen at ticket random from the first 20 integras. The probability that their product is even is.   17/19

11) Out of 30 consecutive integers, 2 or choosen at random. The probability that their sum is odd is..     15/29

12) A bag contains 5 black balls, 4 white balls and three red balls. If  a ball selected randomwise, the probability that it is black or red ball is.       2/3

13) Two dies are thrown simultaneously. The probability of getting a pair of aces is.   1/36

14) An urn contains 9 balls two of which are red, three blue and four black. Three balls are drawn at a random. The probability that they are the same colour is.  5/84

15) A coin is tossed three times. If events A and B are defined as A= two heads come, B= last should be head, then A and B are
A) independent B) dependent.
C) both D) mutually exclusive

16)  5 persons entered the lift cabin on the ground floor of 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, then the probability of all 5 persons leaving at different floors is .      7P5/7⁵

17) A box contains 10 good articles and 6 with defects. One of them is drawn at random. The probability that is either good or has a defect is.      

18) A box contains 6 Nails and 10 nuts. Half of the nails and half of the nuts are rusted. If one item is choosen at random, the probability that is rusted or is a nail is.        11/16

19) A bag contains 5 brown and 4 white socks. A man pulls out two sucks. The probability that these are of the same colour is.   48/108

20) if S is the sample space and P(A)= 1/3 P(B) and A= A U B, where A and B are two mutually exclusive events, then P(A) is.    1/4

21) back on tense X2 white and three black balls and another back why contents for Y2 black balls one bag is selected random and a ball is drawn from it then the probability chosen to be white each two person in the sectors in the throwing a pair of dice the first person through the 9 from both dice will divided the price in the probability the probability that in the year of second century choose and random please please purane setup 100 cards number 1201 card is drawn at random the probability that the number obtained on the card is divisible by 6 or 8 but not by 24 if A and B are two band such that 45 710 then if nbr to events associated to random experiment such that 710 17205 the value is 300 1025 if A and B are two band such that 12 13 14 then equal let a betu even such that 38 58 34 then is equals to 35 to 12:45 0.40.8 0.6 35 12:45 a.m. and such that 0.40.3 0.5 then equals A and B are two even such that then independent there is equals to if A and B are two independent with 35 49 = if A and B are two independent events such that 0.30.5 then a Flashlight has 8 battery is out of 3 are dead if two batteries are selected without replacement and tested the probability that both are dead is a back three balls are drawn at random without replacement then probability of getting exactly One red ball is a bag contains 5 red and 3 Blue Balls three balls are drawn at random without replacement then the probability that exactly two of the three balls are where red the first ball being raid is any college 30% student fail in physics 25% fail in the mathematics and 10% fail in both on student is choose the random the probability that she has failed in mathematics three person ABC fired Sitarganj intern starting with a the probability of heating the target 80.4 0.3 and 0.2 respectively the probability of two sheets is A and B are two students the chances of solving a problems are won by 3 14 respectively if the probability of the making common errors is 120 and they obtained the same answer then the probability of the answers two cards drawn from a well support day of 52 playing cards with replacement the probability that both cards are going a box contains 3 Orange balls three green balls in two Blue Balls three balls are drawn at random from the box without replacement the probability of drawing to Green balls and one blue ball is if two events are independent them they must be much really exclusive the sum of their probability must be equals to 1:00 a.m. two dice are so many if it is known that the sum of the numbers on the dice was a less than 6 the probability of getting the sum 3 age if A and B are such that 5923 if A and B are two event such that then a dice thrown and a card is selected random from a deck of 52 playing card the probability of getting an even number of the day assume that in a family it child is equally likely to be a boy or girl family with 3 children 40.6 then is equals to let a and b two events such that 0.60.2015 then equals to

MISCELLANEOUS-1

1) If an unbiased coin is tossed, what is the probability of obtaining a head ? 1/2

2) Two unbiased coins are tossed. What is the probability of obtaining tail in both the coins? 1/4

3) An unbiased dice is rolled.

a) what is the probability of getting 5 ? 1/6

b) what is the probability of getting an even number ? 1/2

c) what is the probability of getting a number greater than 3 ? 1/2

4) Two unbiased dice are thrown together. Find the probability of obtaining

A) 5 in both the dice. 1/36

B) 3 in one and 4 in other. 1/18

C) same number in both the dice. 1/6

5) A card is drawn at random from a pack of 52 cards. Find the probability of obtaining

A) a black card. 1/2

B) in diamond. 1/4

C) an Ace. 1/13

6) A card is drawn at random from a pack of 52 cards. What is the probability that 

A) The card is king. 1/13

B) the card is not king. 12/13

7) An urn contains 5 red and 4 white balls. Three balls are drawn at random from the URN. What is the probability that all the balls are red ? 5/42

8) A leap year is selected at random. What is the probability that it will contain 53 Mondays? 2/7

9) A box contains 3 white and 5 black balls. A ball is drawn at random. What is the probability that the ball is black ? Also find the odds in favour of the event and against the event. 5/8, 5: 3, 3: 5

10) A card is drawn at random from the pack a pack of 52 cards. Find the probability that the card is a spade or an ace ? 4/13

11) Two unbiased dice are thrown. What is the probability that the sum of the digits on the dice is 7. Find also the odds in favour of this event. 1/6, 1:5 

12) 8 men in a company of 25 are graduates. If 3 men are selected from 25 men at random, what is the probability that

A) they are all graduates. 14/575

B) at least one of them is graduate. 81/115

13) There are 4 white and and 3 black balls in a bag. If 4 balls are drawn at random then what is the probability that 2 of them are white and two are black. 18/35

14) three unbiased coins are tossed. What is the probability that all of their heads. 1/8

15) A card is drawn at random from a pack of 52 cards. What is the probability that

A) the card is not a club ? 3/4

B) The card is neither a spade nor a heart ? 1/2

16) Two unbiased dice are rolled. what is the probability that product of the digits in the dice is 12. 1/9

17) In a family there are two childrens. Find the probability that they will have a different birthdays. 364/365

18) A pair of dice is thrown. Find the probability that sum of the two numbers is in either 8 or 10. 7/9

19) A box contains 6 green balls and 4 yellow balls. three balls are drawn from the box at random. what is the probability that out of three balls 2 are green and one is yellow. 1/2

20) Four unbiased dice are thrown at random. Find the probability after getting different digits in the four dice. 5/18

21) six unbiased coins are tossed together. what is the probability of getting.

A) exactly four heads. 15/64

B) at least four heads. 11/32

22) five unbiased coins are tossed together. Find the probability of getting 3 heads and two tails. 5/16

23) Three unbiased coins are tossed together. Find the sample space in connection with it. find the probability of obtaining.

A) at least one head. 7/8

B) exactly one tail. 3/8

24) 10 balls are distributed at random in boxes. what is the probability of getting 3 balls in the first box? 5120/19683

24) If 20 dates are named at random, what is the probability that 5 of them will be Mondays? (15504x6¹⁵)/7²⁰

25) two unbiased dice are thrown. find the probability that the sum of the faces equals or exceed 10. 1/6

26) If for the two events A and B, P(A)= 3/8, P(B)= 5/8 and P(A U B) = 3/4 then find the value of P(A/B) and P(B/A). also examine whether A and B are independent or not. NO

27) there are two identical urns. One of them contains 4 white and 3 red balls and the other contains 3 white and 7 red balls. An URN is chosen at random and a ball is drawn from it. find the probability that the ball is white. if the ball is white then find the probability that it is taken from the 1st urn. Bay's. 61/140, 40/61

) if A and B are two independent events and P(A)=1/5, P(B)= 2/3 then find the value of P(A U B). 11/15

) if A and B are two events such that P(A)= 1/3, P(B)=1/4 and P(A U B) = 1/2 then find the values of P(A ∩B') and P(A/B'). 1/4,1/3

) there are 4 white, 3 red and 3 blue balls in a box and 5 white, 4 red and 3 blue balls in another box. If a ball is drawn at random from each of the boxes then find the probability that both of them are same colour. 41/120

) there are 3 red and 4 white balls in a bag. Two balls are drawn at random one after another, without replacement.

A) What is the probability that the ball drawn second time is white ?

B) under the condition that the second ball is white; what is the probability that the first ball is white ? 4/7, 1/2

) A and B are two events such that P(A U B)= 7/8, P(A∩ B)= 1/4, P(B')= 1/4. Find the values of P(A), P(B) , P(A ∩ B'). 3/8, 3/4, 1/8

) A box contains 4 red and 3 blue balls. two balls are drawn at time twice from this box. find the probability that the first two balls are red and the next two balls are blue when the first two balls are

A) replaced. 2/49

B) not replaced before drawing the next two. 3/35

) the first bag contains 5 white and 4 Black balls. The second bag contains 3 white and 7 black balls. A ball is drawn at random from the first bag and is kept in the second bag. Now a ball is drawn at random from the second bag. what is the probability that the ball is white ? 32/99

) if P(A)= 2/3, P(B)= 1/2 and P(A U B)= 1 than find the values of P(A/B), P(A/B'), P(A' ∩ B'). Are the events A and B be mutually expensive ? 1/3, 1, 0; no

) In an examination 30% students failed in mathematics, 20% students failed in chemistry and 10% students failed in both mathematics and Chemistry. A students is selected at random.

A) what is the probability that the student may fail in mathematics if it is known that the students failed in chemistry ? 1/2

B) What is the probability that the student may fail in mathematics or Chemistry ? 2/5

) The chance of solving a problem by three students are 2/7, 3/8, 1/2 respectively. If each of them try independently, find the chance that the problem is solved. 87/112

) if A and B be two independent events and P(A)= 2/3, P(B)=3/5 then the value of P(A UB) and P(A ∩ B). 2/5, 13/15

) A candidate is selected for interview for the posts. For the first post there the three candidates, for the second post there are four candidates and for third post there are 2 candidates. What is the chance of his getting at least one post ? 3/4

) There are three identical boxes containing red and blue balls. In the first box there are 3 red and 2 blue balls, in the second there are 4 red and 5 blue balls and in the third there are 2 red and 4 blue balls. A box is chosen at random and a ball drawn from it. If the ball drawn be red then what is the probability that it has been drawn from the second box. Bay's. 10/31

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

) The probability of winning a player is 3/10 if the path of running is fast, and the probability of winning is 2/5 if the path is slow. At a particular day the probability that the path is fast 7/10 and the probability that path is slow is 3/10. find the probability of winning of that player on that day. 33/100