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
7) BAYE'S THEOREM OR INVERSE PROBABILITY:
Let A₁, A₂, ........Aₙ be n mutually exclusive and exhaustive events of the the of the sample space S and A is event which can occur with any of the events then
P(Aᵢ/A) = {P(Aᵢ) P(A/Aᵢ)}/ {ⁿᵣ₌₁∑P(A₁) P(A/A₁)}
7) BINOMIAL DISTRIBUTION FOR REPEATED TRIALS:::
Binomial Experiment:
Any experiment which has only two outcomes is known as binomial experiment.
Outcomes of such an experiment are known as success and failure probability of success is denoted by p and probability of failure by q. p+ q = 1
If binomial experiment is repeated n times, then (p+q)ⁿ =ⁿC₀qⁿ + ⁿC₁pqⁿ⁻¹ + ⁿC₂p²qⁿ⁻² + ....... + ⁿCᵣpʳqⁿ⁻¹ + ....... + ⁿCₙ pⁿ = 1
a) Probability of exactly r successes in n trials =ⁿCᵣpʳqⁿ⁻¹
b) Probability of at most r successes in n trials = ʳₖ₌₀∑ⁿCₖ pᵏ qⁿ⁻ᵏ
c) Probability of at least r success in n trials= ʳₖ₌₀∑ⁿCᵣpᵏqⁿ⁻ᵏ
d) Probability of having 1ˢᵗ success at the rᵗʰ trials= p qʳ⁻¹
The mean the variance and the standard deviation of binomial distribution are np, npq, √(npq) .
9) SOME IMPORTANT RESULTS:
a) Let A and B be two events, then
(i) P(A)+ P(A') = 1
(ii) P(A+B)= 1 - P(A'B')
(iii) P(A/B)= P(AB)/P(B)
(iv) P(A+B)=P(AB)+P(A'B)+P(AB')
(v) A⊂B⇒P(A) ≤ P(B)
(vi) P(A' B)= P(B) - P(AB)
(vii) P(AB)≤ P(A) P(B) ≤ P(A+B) ≤ P(A)+ P(B)
viii) P(AB)= P(A)+ P(B) - P(A+B)
ix) P(exactly one event)= P(A B') + P(A' B)
= P(A)+ P(B) - 2P(AB)
= P(A+B) - P(AB)
x) P(neither A nor B) = P(A' B')
= 1 - P(A+B)
xi) P(A'+ B') = 1 - P(AB)
b) Number of exhaustive cases of tossing n coins simultaneously (or of tossing a coin n times)= 2ⁿ
c) Number of exhaustive cases of throwing n dice simultaneously (or throwing one dies dies n times)= 6ⁿ
d) Playing Cards:
i) Total cards: 52(26 red, 26 black)
ii) Four suits: Heart, Diamond, Spade, Club-- 13 cards each.
iii) Court cards: 12(4 Kings, 4 Queens, 4 jacks)
iv) Honour Cards:16( 4 aces, 4 Kings, 4 Queens, 4 Jacks)
e) Probability Regarding n Letters and their envelopes:
if n letters corresponding to n envelopes placed in the envelopes at random, then
i) Probability that all letters are in right envelopes= 1/n!
ii) Probability that all letters are not in right envelopes= 1- 1/n!
iii) Probability that no letters is in right envelopes = 1/2! - 1/3! + 1/4! - ......+ (-1)ⁿ 1)n!
iv) Probability that exactly r letters are in right envelopes
= 1/r![1/2! - 1/3! + 1/4! - ....+(-1)ⁿ⁻ʳ 1/(n -r)!]
Exercise -A
1) An unbiased dice is thrown. What is the probability of getting:
A) an even number. 1/2
B) a multiple of 3. 1/3
C) an even number or a multiple of 3. 2/3
D) an even number and a multiple of 3. 1/6
E) a number 3 or 4. 1/3
F) an odd number. 1/2
G) A number less than 5. 2/3
H) A number greater than 3. 1/2
I) A number between 3 and 6. 1/3
2) Two unbiased coins are tossed simultaneously. Find the probability of getting:
A) Two heads. 1/4
B) one head. 1/2
C) one tail. 1/2
D) at least one head. 3/4
E) at most one head. 3/4
F) No head. 1/4
3) Three unbiased coins are tossed together. Find the probability of getting:
A) all heads. 1/8
B) two heads . 3/8
C) one head. 3/8
D) at least two heads. 1/2
4) Two dice are thrown simultaneously. Find the probability of getting:
A) an even number as the sum. 1/2
B) the sum as the prime number. 5/12
C) a total atleast 10. 1/6
D) a doublet of even number. 1/12
E) A multiple of 2 on one dice and a multiple of 3 on the other. 11/36
F) same number on the both dice. 1/6
G) a multiple of 3 as the sum. 1/3
5) Find the probability that a leap year selected random, will contain 53 Sundays. 2/7
6) What is the probability that a number selected from the numbers 1,2,3,..... 25 is a prime number, when each of the given numbers is equally likely to be selected. 9/25
7) 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
8) A pack of playing cards consists of 52 cards, each of the 52 cards being equally likely to be drawn. Find the probability that the card drawn is:
A) An ace. 1/13
B) red. 1/2
C) either red or King. 7/13
D) red and a king. 2/26
E) a face card. 4/13
F) a red face card. 2/13
G) 2 of spade. 1/12
H) 10 of a black suit. 1/26
9) The king queen and jack of clubs are removes from a deck of 52 playing cards and the well shuffled. One card is selected from the remaining cards. Find the probability of getting:
A) A heart. 13/49
B) a king. 3/49
C) a club. 10/49
D) the 10 of Hearts. 1/49
10) A bag contains 3 red and 2 blue marbles. A marble is drawn at random of drawing a blue marble. 2/5
11) A bag contains 5 red balls and some blue balls. If the probability of drawing a blue ball is double that of a red ball, Find the number of blue balls in the bag. 10
12) A bag contains 12 balls out of which x are white.
A) If one ball is drawn at random, what is the probability that it will be a white ball. x/12
B) if 6 more white balls are put in the bag, the probability of drawing a white ball will be double than that A. Find x. 3
13) It is know that a box of 600 electric bulbs contains 12 defective bulbs. One bulb is taken out at a random from this box. What is the probability that it is a non defective bulb ? 0.98
14) 17 cards number 1,2,3,....17 are put in a box and mixed thoroughly. One person draws a card from the box. Find the probability the number on the card is :
A) odd. 9/17
B) a prime. 7/17
C) divisible by 3. 5/17
D) divisible by 3 and 2 both. 2/17
15) Cards marked with the numbers 2 to 101 are placed in a box and mixed thoroughly. One card is drawn from this box. Find the probability that the number on the card is:
A) an even number. 1/2
B) a number less than 14. 3/25
C) a number which is a perfect square. 9/100
D) A prime number less than 20. 2/25
16) 1000 tickets of a lottery were sold and there are 5 prizes on these tickets. If Saket has purchased one lottery ticket, what is the probability of winning a prize ? 0.005
17) A child has a block in the shape of a cube with one letter written on each face as shown
A B C D E A
The cube is thrown once. What is the probability getting
A) A. 1/3
B) D. 1/6
18) A bag contains 5 red balls, 8 white balls, 4 green balls and 7 black balls. If one ball is drawn at random, find the probability that it is:
A) black. 7/24
B) red. 5/24
C) not Green. 5/6
19) A die is thrown. Find the probability of getting:
A) a prime number. 1/2
B) two or four. 1/3
C) a multiple of 2 or 3. 2/3
20) In a simultaneous throw of a pair of dice, find the probability of getting:
A) 8 as the sum. 5/36
B) a doublet. 1/6
C) a doublet of prime numbers. 1/12
D) a doublet of odd numbers. 1/12
E) a sum greater than 9. 1/6
F) an even number on first. 1/2
G) an even number on one and a multiple of 3 on the other. 11/36
H) neither 9 nor 11 as the sum of the numbers on the faces. 5/6
I) a sum less than 6. 5/18
J) a sum less than 7. 5/12
K) a sum more than 7. 5/12
21) Three coins are tossed together. Find the probability of getting:
A) exactly two heads. 3/8
B) at least two heads. 1/2
C) at least one head and one tail. 3/4
22) what is the probability that an ordinary year has 53 Sundays ? 1/7
23) what is the probability that a leap year has 53 Sundays and 53 Mondays. 1/7
24) A and B throw a pair of dice. If A throws 9, find B's chance of throwing a higher number. 1/6
25) Two unbiased dice are thrown. Find the probably that the total of the numbers on the dice is greater than 10. 1/12
26) A card is drawn at random from a pack of 52 cards. Find the probably that the card is drawn is:
A) a black King. 1/26
B) either a black card or a king. 7/13
C) black and a king. 1/26
D) a Jack, Queen or a King. 3/13
E) neither a heart nor a king. 9/13
F) spade or an ace. 9/13
G) neither an ace nor or a king. 11/13
27) In a lottery 50 tickets numbered 1 to 50, one ticket is drawn. Find the probability that the drawn ticket bears a prime number. 3/10
28) An urn contains 10 red and 8 white balls. One ball is drawn at random. Find the probability that the ball drawn is white. 4/9
29) A bag contains 3 red balls, 5 black balls and 4 white balls. A ball is drawn at random from the bag. What is the probably the ball drawn is:
A) white ? 1/3
B) red ? 1/4
C) black ? 5/12
D) not red ? 3/4
30) What is the probability that a number selected from the numbers 1, 2, 3,.....15 is a multiple of 4 ? 1/5
31) A bag contains 6 red, 8 black and 4 white balls. A ball is drawn at random. What is the probability that ball drawn is not black ? 5/9
32) A bag contains 5 white and 7 red balls. One ball is drawn at random. What is the probability that ball drawn is white. 5/12
33) Tickets number from 1 to 20 are mixed up and a ticket is drawn at random. What is the probability that the ticket drawn has a number which is a multiple of 3 or 7 ? 2/5
34) In a lottery there are 10 prizes and 25 blanks. What is the probability of getting a prize. 2/7
35) if the probability of winning a game is 0.3, what is the probability of loosing it. 0.7
36) A bag contains 5 black, 7 red and 3 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is:
A) red. 7/15
B) Black or white. 8/15
C) not black. 2/3
37) A black contains 4 red, 5 black and 6 white balls. A ball is drawn from the bag at random. Find the probability that the ball drawn is
A) white. 2/5
B) red. 4/15
C) not black. 2/3
D) red or white. 2/3
38) Fill in the blanks:
a) Probability of a sure event is--
b) probability of an impossible event is ____
c) The probability of an event (other than sure and impossible event) lies between ___
d) Every elementary event associated to a random experiment has____ probability.
Answer: 1, 0, 0 and 1 , equal
Exercise -B
1) Find the probability of getting a head in a toss of an unbiased coin. 1/2
2) In a simultaneous toss of two coins, find the probability of getting:
A) 2 heads. 1/4
B) exactly one head. 1/2
C) exactly 2 tails. 1/4
D) exactly one tail. 1/2
E) no tails. 1/4
3) Three coins are tossed once. Find the probability of getting:
A) all heads. 1/8
B) atleast two heads. 1/2
C) atmost two heads. 7/8
D) no heads. 1/8
E) exactly one tail. 3/8
F) exactly two tails. 3/8
G) a head on first coin.
H) atleast one head and one tail. 3/4
4) A die is thrown. Find the probability of getting:
A) an even number. 1/2
B) a prime number. 1/2
C) a number greater than or equal to 3. 2/3
D) a number less than or equal to 4. 2/3
E) a number more than 6. 0
F) a number less than or equal to 6. 1
G) 2 or 4. 1/3
H) A multiple of 2 or three. 2/3
5). Two dice are thrown simultaneously. Find the probability of getting:
A) an even number as the sum. 1/2
B) the sum as a prime number. 5/12
C) a total of atleast 10. 1/6
D) a doublet of even number. 1/12
E) a multiple of 2 on one dice and a multiple of 3 on the other dice. 11/36
F) same number on both dice. 1/6
G) a multiple of 3 as the sum. 1/3
H) neither a doublet nor a total of 8 will appear. 13/18
I) the sum of the numbers obtained on the two dice is neither a multiple of 2 nor a multiple of 3. 1/3
6) Three dice are thrown together. Find the probability of getting:
A) a total of atleast 6. 103/108
B) a total of 17 or 18. 1/54
7) What is the probability that a number selected from the numbers 1, 2, 3,...., 25, is prime number, when each of the given numbers is equally likely to be selected? 9/25
8) 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
9) A coin is tossed. If head comes up, a die is thrown but if tail comes up, the coin is tossed again. Find the probability of getting:
A) two tails. 1/8
B) head and number 6. 1/8
C) head and an even number. 3/8
10) A letter is chosen at random from the word ASSASSINATION. Find the probability that letter is
A) a vowel. 6/13
B) a consonant. 7/13
11) 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 prize. What is the probability of winning the prize in the game? 1/38760
12) On her vacations Ram visits four cities A, B, C , D in a random order. What is the probability that he visits.
A) A before B. 1/2
B) A before B and B before C. 1/6
C) A first and B last. 1/12
D) A either first or second. 1/2
E) A just before B. 1/4
13) 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). 1/2
B) P(1 or 3). 1/2
C) P(not 3). 5/6
14) If 4-digit numbers greater than or equal to 5000 are randomly formed from the digits 0, 1, 3, 4 and 7, what is the probability of forming number divisibile by 5 when
A) the digits may be repeated. 2/5
B) the repetation of digits is not allowed. 3/8
15) One card is drawn from a pack of 52 playing cards, each of the 52 cards being equally likely to be drawn. Find the probability that the card drawn is:
A) an ace. 1/13
B) red . 1/2
C) either red or king. 7/13
D) red and a king. 1/26
16) An urn contains 9 red, 7 white and 4 black balls. If two balls are drawn at random, find the probability that:
A) both the balls are red. 18/95
B) one ball is white. 91/190
C) the ball are of the same colour. 63/190
D) one is white and other red. 63/190
17) A bag contains 6 red, 4 white and 8 blue balls. If three balls are drawn at random, find the probability that
A) one is red, one is white and one is blue. 4/17
B) one is red and two are white. 3/68
C) two are blue and one is red. 7/34
D) one is red. 33/68
18) A bag contains 4 red, 7 white and 5 black balls. If two balls are drawn at random, find the probability that
A) both the balls are white. 7/40
B) one is black and the other red. 1/6
C) both the balls are of the same colour. 37/120
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. 20C5/60C5
B) at least one will green? 1 - 30C5/69C5
20) A box contains 8 red, 3 white and 9 blue balls. 3 balls are drawn at random, what is the probability that
A) all the three balls are blue balls. 7/95
B) all the balls are of different colours. 18/95
21) A box contains 5 red marbles, 6 white marbles and 7 black marbles. 2 marbles are drawn from the box, what is the probability that both the balls are red or both are black. 31/153
22) 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) 1 ticket. 999/1000
B) two tickets. 9990C2/10000C2
C) 10 tickets. 9990C10/10000C10
23) The number of lock of a suitcase has 4 wheels, each labelled with ten 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. 1/5040
24) Out of 100 students, two sections of 40 and 60 students are formed. If you and your friends are among the 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
25) Four cards are drawn at random from a pack of 52 playing cards. Find the probability of getting:
A) all the four cards of the same suit. 198/20825
B) all the four cards of the same number. 13/270725
C) one card from each suit. 2197/20825
D) two red cards and two black cards. (26C2 x 26C2)/52C4.
E) all cards of the same colour. 2.26C4/52C4.
F) all face cards. 12C4/52C4
26) In a lottery of 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 drawn has prime number. 17/35
C) one ticket has prime number. 3/7
27) In a lottery, a person chooses six different numbers at random from 1 to 20, and if these six numbers match with six numbers already fixed by the lottery committee, he wins the prize. What is the probability of winning the prize in the game? 1/38760
28) A word consists of 9 letters, 5 consonants and 4 vowels. Three letters are chosen at random. What is the probability that more than one vowel be selected. 17/42
29) Four persons are to be selected at random from a group of 3 men, 2 women and 4 children. Find the probability of selecting:
A) 1 man, 1 woman, 2 children. 2/7
B) exactly 2 children. 10/21
C) 2 women. 1/6
30) A box contains 10 bulbs, of which just three are defective. If a random sample of five bulbs is drawn, find the probabilities that the sample contains:
A) exactly one defective. 5/12
B) exactly two defective. 5/12
C) no defective bulbs. 1/12
31) 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) atleast one is defective.
D) none is defective.
32) 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. 88/203
33) Twelve balls are distributed among three boxes, find the probability that the first box contains three balls. (12C3 x 2⁹)/3¹²
34) Five marbles are drawn from a bag which contains 7 blue and 4 black marbles. Find the probability that:
A) all will be blue. 1/22
B) 3 will be blue and 2 black. 5/11
35) Find the probability that when a hand of 7 cards is dealt from a well shuffled deck of 52 cards. It contains
A) all 4 kings. 1/7735
B) exactly 3 kings. 9/1547
C) atleast 3 kings. 46/7735
36) In a single throw a three dice, determine the probability of getting:
A) a total of 5. 1/36
B) a total of atmost 5. 5/108
C) a total of atleast 5. 53/54
37) Three dice are thrown simultaneously. Find the probability that:
A) all of them show the same face. 1/36
B) all show distinct faces. 5/9
C) two of them show the same face. 5/12
38) What is the probability that in a group of
A) 2 people both will have the same birthday? 1/365
B) 3 people, atleast two will have the same birthday. (364x 363)/365²
39) The letters of the word SOCIETY are placed at random in a row. What is the probability that three vowels come together. 1/7
40) The letters of the word SOCIAL are placed at random in a row. What is the probability that the vowels come together. 1/5
41) The letters of the word CLIFTON are placed at random in a row. What is the probability that the vowels come together. 2/7
42) The letters of the word FORTUNATES are placed at random in a row. What is the probability that the two T come together.
43) The letters of the word UNIVERSITY are placed at random in a row. What is the probability that two I do not come together. 4/5
44) Find the probability that in a random arrangement of the letters of the word UNIVERSITY the two I's come together. 1/5
45) A five digits number is formed by the digits 1,2,3,4,5 without repetition. Find the probability that the number is divisible by 4. 1/5
46) Out of 9 outstanding students in a college, there are 4 boys and 5 girls. A team of four students is to be selected for a quiz programme. Find the probability that two are boys and two are girls. 10/21
47) There are 4 letters and 4 addressed envelopes. Find the probability that all the letters are not dispatched in right envelopes. 23/24
48) The odds in favour of an event are 3:5. Find the probability of occurrence of this event. 3/8
49) The odds in favour of an event are 2:3. Find the probability of occurrence of this event. 2/5
50) The odds in against of an event are 7:9. Find the probability of non-occurrence of this event. 7/16
51) 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 bet. 9:4
52) Two dice are thrown. Find the odds in favour of getting the sum
A) 4. 1:11
B) 5. 1:8
C) What are the odds against getting the sum 6? 31:5
53) What are the odds in favour of getting a spade if the card drawn from a pack of cards ? What are the odds in favour of getting a king ? 1:3, 1:12
54) A fair coin with 1 marked on one face and on the other and a fair die are both tossed, find the probability that the sum of numbers that turns up is
A) 3. 1/12
B) 12. 1/12
55) In a relay race there are five teams A, B, C, D and E.
A) what is the probability that A, B and C finish first, second and third respectively. 1/60
B) what is the probability that A, B and C are first three to finish (in order). 1/19
56) In shuffling a pack of 52 playing cards, four are accidently dropped; find the probability that the missing cards should be one from each suit. 2197/20825
57) Five cards are drawn from a pack of 52 cards. What is the probability that these 5 will contain
A) just one ace. 3243/10829
B) atleast one ace.
58) If a letter is chosen at random from the English alphabet, find the probability that the letters.
A) a vowel. 5/26
B) a constant. 21/26
59) A class consists of 10 boys and 8 girls. Three students are selected at random. What is the probability that the selected group has
A) all boys. 5/34
B) all girls. 7/102
C) 1 boy and 2 girls. 35/102
D) atleast one girl. 29/34
E) at most one girl. 10/17
Exercise - C
Formula:
* (Addition theorem for two events) If A and B are two events associated with a random experiment, then
1) P(A∪B) = P(A) + P(B) - P(A∩B)
* If A and B are mutually exclusive events, then
P(A∩B) = 0
So, P(A UB) = P(A) + P(B)
This is the addition theorem for mutually exclusive events.
2) (Addition Theorem for three events) If A, B, C are three events associated with a random experiment, then
P(A∪B∪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∩B)= P(B∩C) = P(A∩C) = P(A∩B∩C) = 0.
P(A U BU C)= PA) + P(B)+ P(C).
This is the addition theorem for three mutually exclusive events.
3) i) P(A'∩B)= P(B) + P(A∩B)
ii) P(A∩B') =P(A) - P(A∩B)
iii) P(A∩B') U P(A'∩B) = P(A) + P(B) - 2P(A∩B)
* P(A'∩B) is known as the probability of occurrence of B only.
* P(A∩B') is known as the probability of occurrence of A only.
* P(A∩B')U P(A'∩B) is known as the probability of occurrence of exactly one of two events A and B.
* If A and B are two events associated to a random experiment such that A ⊂ B, then A' ∩ B ≠ ∅
4) For any two events A and B
P(A ∩ B) ≤ P(A) ≤ P(A U B) ≤ P(A) + P(B).
5) P(A) + P(B) - P(A UB)) = P(A U B) - P(A ∩B).
6) P(A' ∩ B') = 1 - P(A U B)
+++++++++++++++++++++++++++
1) Given P(A)= 3/5 and P(B) = 1/5, Find P(A or B), if A and B are mutually exclusive events. 4/5
2) A and B are two mutually exclusive events of an experiment. If P(not A)= 0.65, P(A UB)= 0.65 and P(B)= p, find the value of p. 0.30
3) Given P(A)= 1/4 and P(B) = 2/5, P(A U B) = 1/2, find
A) P(A∩ B). 3/20
B) P(A∩ B'). 1/10
if A and B are mutually exclusive events.
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) If A and B are two events associated with a random experiment such that P(A)= 0.3, P(B) = 0.4, P(A U B) =0.5, find
A) P(A∩ B). 0.2
6) If A and B are two events associated with a random experiment such that P(A)= 0.5, P(B) = 0.3,P(A∩ B)=0.2, find
A) P(A U B). 0.6
7) If A and B are two events associated with a random experiment such that P(AU B)= 0.8, P(A ∩B) = 0.3, P(A')= 0.5,
find P(B). 0.6
8) Given P(A)= 1/2 and P(B) = 1/3, Find P(A or B), if A and B are mutually exclusive events. 5/6
9) Given P(A)= 0.4 and P(B) = 0.5, if A and B are mutually exclusive events associated with a random experiment. Then find
A) P(AU B). 0.9
B) P(A' ∩ B'). 0.1
C) P(A' ∩ B). 0.5
D) P(A ∩ B'). 0.4
10) A and B are two events such that Given P(A)= 0.54 and P(B) = 0.69, P(A ∩ B) = 0.35. find
A) P(AU B). 0.88
B) P(A' ∩ B'). 0.12
C) P(A ∩ B'). 0.19
D) P(B ∩ A'). 0.34
11) Fill in the blanks:
P(A) P(B) P(A ∩ B) P(AU B)
A) 1/3 1/5 1/15 ____
B) 0.35 ___ 0.25 0.6
C) 0.5 0.35 ___ 0.7
12) 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(B)= 0.5 P(B)= 0.4, P(A U B)= 0.85.
13) Events E and F are such that P(not E or not F)= 0.25, State whether E and F are mutually exclusive.
14) A, B, C are three mutually exclusive events associated with a random experiment. Given P(B)= 3/2 P(A), P(C)= 1/2 P(B),
find P(A). 4/13
15) A, B, C are events 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(A U B UC)≥ 0.75, then show that P(B ∩ C) lies in the interval (0.23, 0.48).
16) The probability of two events A and B are 0.25 and 0.50 respectively. The probability of their simultaneously occurance is 0.14. find the probability that neither A nor B occurs.
17) There are three events A, B and 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
18) In a race, the odds in favour of horses A, B, C , ad are 1:3,1:4,1:5,1:6 respectively. Find probability that one of them wins the race. 319/420
19) in an easy competition, the odds in favour of competition P, Q, R, S are 1:2,1:3,1:4,1:5 respectively. Find the probability that one of them wins the competition. 114/120
20) 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
21) A card is drawn from a deck of 52 cards. Find the probability of getting an ace or a spade card. 4/13
22) A card is drawn from a deck of 52 cards. Find the probability of getting spade or a king. 4/13
23) Four cards are drawn from a deck of 52 cards. Find the probability that all the drawn cards are of the same colour. 92/883
24) Two cards is drawn from a deck of 52 cards. Find the probability that either both are black or both are kings. 55/211
25) Two card are drawn from a deck of 52 cards. Find the probability that 2 cards drawn are either aces or black cards. 55/21
26) 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
27) Four cards are drawn at a time from a pack of 52 cards. Find the probability of getting all the four cards of the same suit. 44/4165
28) Two cards are drawn from a pack of 52 cards. What is the probability that either both are red or both are kings. 55/221
29) In a single throw of two dice, find the probability that neither a doublet nor a total of 9 will occur. 13/1
30) A die is thrown twice. What is the probability that atleast one of the two throws come up with the number 3 ? 11/36
31) Find the probability of getting an even number on the first die or a total of 8 in a single throw of two dice. 5/9
32) A die is thrown twice. What is the probability that atleast one of the two throws comes up with the number 4 ? 11/36
33) Two dice are thrown together. What is the probability that the sum of the numbers on the two faces is neither divisibile by 3 nor by 4 ? 4/9
34) 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
35) A die has two faces with number '1' three faces each with number '2' and one face with number '3'. If the die is rolled once, determine
A) P(1). 1/3
B) P(1 or 3). 1/2
C) P(not 3). 5/6
36) A natural number is choosen at random from amongst first 500. What is the probability that the number so chosen is divisible by 3 or 5? 233/500
37) One number is chosen from numbers 1 to 100. Find the probability that it is divisible by 4 or 6. 33/100
38) 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
39) 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? 33/50
40) Find the probability of at most two tails or atleast two heads in a toss of three coins. 7/8
41) One number is chosen from numbers 1 to 200. Find the probability that it is divisible by 4 or 6? 67/200
42) 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 atleast one examination. 4/5
43) 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 drawn is either white or red ? 8/13
44) 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 atleast one ball of each colour. 48/91
45) 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
46) A box contains 30 bolts and 40 nuts. Half of 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.
47) 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 ? 5/8
48) Find the probability of getting 2 or 3 tails when a coin is tossed four times. 5/8
49) In an entrance test that is graded on the basis of two examinations, the probability of a randomly selected student passing the first examination is 0.8 and the probability of passing the second examination o.7. the probability of passing atleast one of them is 0.95. What is the probability of passing both. 0.55
50) The probability that a student will pass the final examination in both English and Hindi is 0.5 and the probability of passing neither is 0.1. if the probability of passing the Hindi examination. 0.65
51) A basket contains 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
52) The probability that a person will get an electric contact is 2/5 and the probability that he will not get plumbing contract is 4/7. If the probability of getting atleast one contract is 2/3, what is the probability that he will get both ? 17/105
53) The probability that a student will receive A, B, C or D grade are 0.40, 0.35, 0.15 and 0.10 respectively. Find the probability that a student will receive
A) B or C grade. 0.50
B) at most C grade. 0.25
54) Let A, B and C be three events. If the probability of occuring 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.
55) The probability that a patient visiting a dentist will have a tooth extracted is 0.6, the probability that he will have 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? 0.23
56) The probability that a patient visiting a dentist will have a tooth cleaned is 0.44, the probability that he will have a cavity filled is 0.24 and the probability that he will have a tooth cleaned as well as cavity filled is 0.60. what is the probability that a patient has either a tooth cleaned or a cavity filled? 0.08
57) Probability that Ram 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 Ram will pass in atleast one of these subjects? 31/36
58) In a town of 6000 people 1200 are over 50 years old and 2000 are female. It is known that 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. 13/30
59) Two students Ram and Shyam appeared in an examination. The probability that Ram will qualify the examination is 0.05 and that of shyam will qualify the examination is 0.10. the probability that both will qualify the examination is 0.02. find the probability that:
A) both Ram and Shyam will not qualify the exam. 0.87
B) atleast one of them will not qualify. 0.98
C) only one of them will qualify the exam. 0.11
60) In class XII 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. 3/5
61) In a class of 60 students 30 played football, 32 played cricket and 24 played both football and cricket. If one of these students is selected at random, find the probability that:
A) the student played for football or cricket. 19/30
B) the student has played neither football nor cricket. 11/30
C) the student has played football but not cricket. 2/15
CONDITIONAL 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 occurred and P(B)≠ 0, is called the conditional probability and it is denoted by P(A/B). Thus,
P(A/B)= probability of occurrence of A given that B has already occurred.
Similarly,
P(B/A) when P(A) ≠ 0 is defined as the probability of occurrence of event B when A has already occurred.
The meanings 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
Example:1
Let there be a bag containing 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.
P(B/A)= Probability of drawing a red ball in second draw given that a white ball has already been drawn in the first draw.
P(B/A)= Probability of drawing a red ball from a bag containing 4 white and 4 red balls.
P(B/A)= 4/8= 1/2
Here P(A/B) is not meaningful because A can not occur after the occurrence of event B.
Example: 2
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= {(2,6),(6,2),(3,5),(5,3),(4,4)}
B= There is an even number on first die={(2,1),(2,2),(2,3),(2,4),(2,5),(2,6),(4,1),(4,2),(4,3),(4,4),(4,5),(4,6),(6,1),(6,2),(6,3),(6,4),(6,5)(6,6)}.
P(A/B)= Probability of occurrence of A when B occurs
OR
Probability of occurrence A when B is taken as the sample space.
P(A/B)=(Number of elementary events in B which are favorable to A)/(Number of elementary events in B)
OR
P(A/B)=(Number of elementary events favorable to (A∩B)/(Number of elementary favourable to B) = 3/18
Similarly
P(B/A)= Probability of occurrence of B when A occurs.
OR
Probability of occurrence B when A is taken as the sample space.
P(B/A)=(Number of elementary events in A which are favorable to B)/(Number of elementary events in A)
OR
P(B/A)=(Number of elementary events favorable to (A∩B)/(Number of elementary favourable to A) = 3/5
Example 3:
A die 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 atleast once?
Consider the following events:
A= Number 4 appears atleast once.
B= sum of the numbers appearing is 6.
Required probability=
P(A/B) = Probability of occurrence of A when B is taken as the sample space.
OR
P(A/B)=(Number of elementary events favorable to A which are favourable to B)/(Number of elementary events in B)
OR
P(A/B)=(Number of elementary events favorable to (A∩B)/(Number of elementary favourable to B) = 2/5
OR
In short
P(A/B)= (Number of elementary events favorable to (A∩B)/(Number of elementary favourable to B)
OR
P(A/B)= n(A∩B)/n(B)
= n(A∩B)/n(S)/n(B)/n(S)
OR
P(A/B)= n(A∩B)/n(B)
Similarly
P(B/A)= P(A∩B)/P(A)
Formula:
1) P(A)= P(A∩B)+ P(A∩B')
2) P(B)= P(A∩B)+ P(A'∩B)
3) P(A U B) = P(A∩B)+ P(A∩B') + P(A'∩B).
4) P(A/B)= n(A∩B)/n(B)
5) P(B/A)= n(A∩B)/n(A)
EXERCISE - D
1) A four 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 UB/C). 3/4
F) P(A∩B/C). 1/4
2) A coin is tossed three times. Find P(E/F) in each of the following:
i) E= Head on third toss, F= Heads on first two tosses. 1/2
ii) E= Atleast two heads, F= atmost two heads. 3/7
iii) At most two tails, F= atleast one tail. 6/7
3) Two coins are tossed once. Find P(E/F) in each of the following:
i) E= Tail appears on one coin, F= One coin shows head. 1
ii) 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 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 = nul set. 0
6) 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
7) 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
8) Evaluate P(A U B), If 2P(A)= P(B) = 5/13 and P(A/B)= 2/5. 11/26
9) If P(B)= 0.5 and P(A∩B) = 0.32 find P(A/B). 16/25
10) 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
11) If A and B are two events such that P(A)= 1/3, P(B)= 1/5 and P(A UB)= 11/30. Find
A) P(A/B). 5/6
B) P(B/A). 1/2
12) A basket contains 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
13) 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 atleast one contract is 2/3, what is the probability that he will get both ? 17/105
14) Ten cards numbered 1 through 10 are placed in a box, mixed up thoroughly and then one card is drawn and 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
15) A pair of dice is thrown. If the two numbers appearing on them are different. Find the probability
A) the sum of the numbers is 6. 2/15
B) the sum of the numbers is 4 or less. 1/15
C) the sum of the numbers is 4. 2/15
16) A dice thrown three times, find the probability that 4 appears on the third toss if it is given that 6 and 5 appear respectively on first two tosses. 1/6
17) A die is thrown three times. Events A and B are defined as below:
A= Getting 4 on third die,
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
18) 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
19) A die is thrown three times. If the first throw is a four, find the probability of getting 15 as the sum. 1/18
20) A coin tossed three times, if head occurs on first two tossed, find the probability of getting head on third toss. 1/2
21) A black and a red dice are rolled in order. Find the conditional probability of getting
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
22) 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 atleast one die shows a three. 0
23) 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. 1/3
B) atleast one is a girl. 1/3
24) Given that the two numbers appearing on throwing two dice are different. Find the probability of the event 'the sum of numbers on the dice is 4'. 1/15
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.
B) females, if it is known that the elder child is the female. 1/3, 1/2
26) 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 the children are girls, if it is known that the older child is a girl. 1/2
C) both the children are boys, if it is known that atleast one of the children is a boy. 1/3
27) In a school, there are 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 selected randomly studies in class XII given that the chosen student is a girl. 1/10
28) 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
FORMULA:
If A and B are two events associated with a random experiment, then
1) 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
2) If A, B, C are three events associated with a random experiment, then
P(A∩B∩C)= P(A) P(B/A) P(C/A∩ B).
3) P(A UB)= 1 - P(A') P(B')
Exercise - E
1) From a pack of 52 cards, two are drawn one by one without replacement. Find the probability that the 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) From a deck of cards, three cards are drawn one by one without replacement. Find the probability that each time it is a card of spade. 11/850
4) 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
C) the first is a heart and second is red. 25/204
5) A card is drawn from a well shuffled deck 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
6) Three cards drawn successfully, without replacement from a pack of 52 well shuffled cards. What is the probability that first two cards are kings and third card drawn is an ace ? 2/5525
7) 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
8) Find the chance of drawing two white balls in succession from a bag containing 5 red and 7 white balls, the balls first drawn not being replaced. 7/22
9) A bag contains 5 white, 7 red and 8 black balls. If balls are drawn one by one without replacement, Find the probability of getting all white balls. 1/969
10) A bag contains 5 white and 8 black balls. Two successive drawings of three balls are a time are made 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
11) 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
12) 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
13) An urn contains 10 black and 5 white balls. Two balls at drawn from the urn one after the other without replacement. What is the probability that both drawn balls are black ? 3/7
14) A bag contains 4 white, 7 black and 5 red balls. 3 balls are drawn one after the other without replacement. Find the probability that the balls drawn are white, black and red respectively. 1/24
15) 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
16) A bag contains 19 numbered from 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
17) 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
18) A bag contains 25 tickets, numbered from 1 to 25. A ticket is drawn and then another ticket is drawn without replacement. Find the probability that both tickets will show even numbers. 11/50
19) 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 second an odd number. 5/19
20) A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three orange 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
21) To test the quality of electric bulbs produced in a factory, two bulbs are randomly selected from a larger 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
Exercise - F
1) If P(A)= 7/13, P(B)= 9/13 and P(A∩B)= 4/13, find P(A/B). 4/9
2) If A and B are 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(A∩B)= 0.32 and PB =0.5, find P(A/B). 0.64
4) If P(A)=0.4 , P(B)=0.8, P(B/A)=0.6
Find
A) P(A/B). 0.3
B) P(A U B). 0.96
5) If A and B are two events such that
A) P()=1/3, P(B)=1/4 and P(AUB)=5/12 find
a) P(A/B). 2/3
b) P(B/A). 1/2
B) P(A)==6/11, P(B)= 5/11 and P(AUB)=7/11, find
a) P(A∩B). 4/11
b) P(A/B). 4/5
c) 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
a) P(A/B). 3/4
b) P(B/A). 1/2
c) P(A'/B). 1/4
d) P(A'/B'). 5/8
6) If A and B are two events such that 2P(A)=P(B)= 5/13 and P(A/B)= 2/5, find P(AUB). 11/26
7) If P(A)=6/11, P(B)=5/11 and P(AUB)=7/11, find
A) P(A∩B). 4/11
B) P(A/B). 4/5
C) P(B/A). 2/3
8) 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) P(A/B). 1/2
B) P(B/A). 3/5
9) If A and B are two events such that P(A)=0.3, P(B/A)=0.5 Find
A) P(AUB). 0.75
B) P(A/B). 1/4
10) If P(not A)=0.7, P(B)=0.7 and P(B/A)= 0.5, then find
A) P(AUB). 0.85
a) P(A/B). 3/14
11) 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
a) P(A/B). 0.64
b) P(A U B). 0.98
12) A fair die is thrown. Consider the events
A= {1,3,5}, B={2,3}, 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(B/C). 1/2
E) P(AUB/C). 3/4
F) P(A∩B/C). 1/4
13) Three events A, B and C have Probabilities 2/5, 1/3 and 1/2 respectively. Given that P(A∩C)= 1/5 and P(B∩C)= 1/4, find the values of
A) P(C/B). 3/4
B) P(A'∩C'). 3/10
14) If P(A)= 3/8, P(B)= 1/2 and P(A∩B)=1/4 find
A) P(A'/B'). 3/4
B) P(B'/A'). 3/5
15) A coin is tossed three times. Find P(A/B) in each case of the following:
a) A= Head on 3rd toss, B= heads on first two tosses. 1/2
b) A= at least two heads, B= atmost two heads. 3/7
c) A= at most two tails, B= at least one tail. 6/7
16) 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, 'at most 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
17) Two coins tossed once. Find P(A/B) in each case of the following:
A) tail appears on one coin, B= One coin shows head. 1
B) A= no tail appears, B= No head appears. 0
18) Two coins are tossed. What is the probability of coming up two heads if it is known that atleast one head comes up. 1/3
19) A die is thrown three times. Find P(A/B) and P(B/A), if
A) 4 appears on the third toss, B= 6 and 5 appear respectively on 1st two tosses. 1/6, 1/36
20) A dice is thrown twice and the sum of the numbers appearing is observed to be six. What is the condition of probability that the number 4 has appeared atleast once. 2/5
21) A dice is thrown. If outcome is an odd number, what is the probability that it is prime. 2/3
22) A dice thrown twice and the sum of the numbers appearing is observed to be 8, what is the conditional probability that the number 5 has appeared at least once. 2/5
23) 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 appeared at least once. 1/3
24) A die is thrown three times. Events A and B are defined as follows:
A: 4 on third throw, B: 6 on the first and 5 on the second throw, find the probability of A given that B has already occurred. 1/6
25) two dice are thrown. Find the probability that the number appeared has the sum 8, if it is known that the second die always exhibits 4. 1/6
26) A pair of dice is thrown. Find the probability of getting 7 and the sum, If it is known that second dice always exhibits an odd number. 1/6
27) 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 a Prime number. 1/6
28) A pair of dice is thrown. Find the probability of getting the sum 8 or more, if 4 appears on the first die. 1/2
29) 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
30) A pair of dice is thrown. Let E be the event that the sum is greater than or equal to 10 and F be the event 5 appears on the first die. Find P(E/F). If is the event 5 appears on at least one die. find P(E/F). 1/3, 3/11
31) 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 six. 1/10
32) A black and a red die rolled.
A) find the conditional probability of getting a sum greater than 9, given that the black die resulted in a 5. 1/3
B) Find the conditional probability of getting the sum of 8, given that the red die resulted in a number less than 4. 1/9
33) Find the probability the sum of the numbers is showing on two dies is 8, given that at least one die does not show 5. 3/25
34) two numbers are selected at random from Integers 1 through 9. If the sum is even, find the probability that both the numbers are odd. 5/8
35) 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
36) 10 cards numbered 1 to 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 a more than three. What is the probability that it is an even number ? 4/7
37) the probability that a student selected at random from a class will pass in the Mathematics is 4/5, and the probably that/he/she is 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
38) The probability that a certain person 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 buys a shirt. 0.12, 0.6
39) 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 probably that a student selected randomly studies in class XII given that chosen student is a girl ? 1/10
40) Assume that is each born child 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 girl. 1/3
41) A couple has 2 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
42) Mother father and son line up at random for a family picture. If A and B are two events given by A= son on one end, B= father in the middle, find
A) P(A/B). 1
B) P(B/A). 1/2
43) 10% of the bulbs produced in a factory are red colour and 2% are red and defective. If one bulb is picked at random, determine the probability of its being defective if it is red. 1/5
44) 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
45) 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 the selected bag a ball is drawn. Let A be the event that the first bag is selected, B be the event that the second bag is 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(CB). 2/3
46) Three distinguishable balls are distributed in three cells. Find the conditional probability that all the three occupy the same cell, given that atleast two of them are in the same cell. 1/7
47) A coin is tossed, then a die is thrown. Find the probability of getting a 6 given that head came up. 1/6
48) Consider the experiment of tossing a coin. If the coin shows head loss it again but if it shows tail then throw a dice. Find the conditional probability of the event the die shows a number greater than 4, given that there is atleast one tail. 2/9 or 2/7
49) 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 atleast one die shows a 2. 3/8 or 1/4
50) A committee of 4 students is selected at random from a group consisting of 8 boys and 4 girls. Given that there is atleast one girl in the committee, calculate the probability that there are exactly 2 girls in the committee. 168/425
51) 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. Find
A) the probability that she reads neither Hindi nor English newspapers. 1/5
B) If she reads Hindi newspaper, find the probability that she reads English newspapers. 1/3
C) if she reads English newspaper, find the probability that she reads Hindi newspaper. 1/2
52) An electronic assembly consists of two sub-systems say A and B. From previous testing procedures, the following probabilities are assumed to be known.
P(A fails)= 0.2, P(B fails alone)= 0.15, P(A and B fail)= 0.15
Evaluate the following probabilities:
A) P(A fails/B has failed). 1/2
B) P(A fails alone). 0.05
____________
Formula:
1) If A and B are independent events associated with a random experiment, then
P(A∩B)=P(A) P(B).
2) If A and B are independent events associated with a random experiment, then
i) A' and B are independent.
ii) A and B ' are independent events.
iii) A' and B ' are also independent events.
Exercise - G
1) A coin is tossed thrice and all the eight outcomes are assumed equally likely. In which of the following cases are the following events A and B are independent?
i) A= the First throw results in head,
B= the last throw results in tail. Y
ii) A= the number of heads is odd,
B= the number of tails is odd.
iii) A= the number of heads is two
B= the last throw results in head.
2) A lot contains 50 defective and 50 non-defective bulbs. Two bulbs are drawn random, one at a time, with replacement. The events A, B, and C are defined as
A: the first bulb is defective
B: the second bulb is non defective.
C: the two bulbs are both defective or both non defective.
Determine whether
i) A, B and C are pairwise independent.
ii) A, B and C are mutually independent.
3) A card is drawn from a pack of 52 cards so that each card is equally likely to be selected. In which of the following cases are the events A and B independent?
i) A= the card drawn is a king or queen,
B= the card drawn is a queen or Jack.
ii) A= the card drawn is black,
B= the card drawn is a king. Y
iii) B= the card drawn is a spade,
B= the card drawn in an ace. X
4) A coin is tossed thrice and all eight outcomes are equally likely.
E: The first throw results in head.
F: The last throw results in tail.
Prove that events E and F are independent.
5) A coin is tossed three times. Let the events A, B and C be defined as follows:
A= first toss is head, B= second toss his head, C= exactly two heads are tossed in a row.
Check the independence of
i) A and B . I
ii) B and C. D
iii) C and A. I
6) an unbiased dice is thrown twice. Let the event A be odd number on the first throw and B the event odd number on the second throw. Check the independent of events A and B.
7) If A and B be two events such that P(A)= 1/4, P(B)= 1/3 and P(A U B)= 1/2, show that A and B are independent events.
8) Two dice are thrown together and the total score is noted. The events E, F and G are a total 4, a total of 9 or more, and a total divisible by 5, respectively. Calculate P(E), P(F) and P(G) and decide which pair of events, if any, are independent.
9) Three coins are tossed. consider the event:
E= 3 heads or 3 tails,
F= at least two heads
G= Atmost two heads.
of the pairs (E, F), (E,G) and (F,G) which are independent ? Which are dependent ?
10) A fair coin and an unbiased die are tossed. Let A be the event 'head appayers on the coin' and B be the event '3 on the die' check whether A and B are Independent or not. Y
11) A die is marked 1, 2, 3 in red and 4, 5 ,6 in green is tossed. Let A be the event 'nunber is even' and B be the event 'numer is red. Are A and B independent. N
12) Event A and B are such that P(A)= 1/2, P(B)=7/12 and P( not A or not B)= 1/4. State whether A and B are Independent? N
13) A die if thrown once. If A is the event 'the number appearing is a multiple of 3' and B is the event ' the number appearing is even'. Are the events A and B independent ? Y
14) Two dies are thrown together. let A be the event 'getting 6 on the first die' and B be the event 'getting 2 on the second die'. Are the events A and B independent ? Y
15) For a loaded die, the probabilities of outcomes are given as under:
P(1)=P(2)= 2/10, P(3)= P(5)= P(6) = 1/10 and P(4)= 3/10.
The die is thrown two times. Let A and B be the events as defined below
A= getting same number each time,
B= getting a total score of 10 or more.
Determine whether or not A and B are independent events
16) In the above question, if the die were fair, determine whether or not the events A and B are independent. N
17) In the two dice experiment, if A is the event of getting the sum of the numbers on dice as 11 and B is the event of getting a number other than 5 on the first die, find
P(A and B) are A and B independent events? N
18) Given two independent events A and B such that P(A)= 0.3 and P(B)= 0.6, find
A) P(A∩B). 0.18
B) P(A∩B'). 0.12
C) P(A'∩B). 0.42
D) P(A'∩B'). 0.28
E) P(AUB). 0.72
F) P(A/B). 0.3
G) P(B/A). 0.6
19) If P(not B)= 0.65,P(AUB)= 0.85 and A and B are independ events, then find P(A). 0.77
20) if A and B are two Independent events such that P(A'∩B)=2/15 and P(A∩B')= 1/6, then find P(B). 1/6 or 4/5
21) A and B are two Independent events. The probability that A and B occur is 1/6 and the probability that neither of them occurs is 1/3. Find the probability of a occurrence of two events. 1/3 , 1/2 or 1/2, 1/3
22) If A and B are two independent events such that P(AUB)=0.60 and P(A)=0.2, find P(B). 0.5
23) Given that the events A and B are such that P(A)=1/2, P(AUB)= 3/5 and P(B)= p. find p, if they are
A) mutually exclusive. 1/10
B) independent . 1/5
24) if A and B are two events such that P(A)= 1/4, P(B)= 1/2 and P(A∩B)=1/8, find P(not A and not B). 3/8
25) events E and F are independent find
A) P(F), if P(E)= 0.35 and P(E F)= 0.6 . 5/13
26) P(A)= 0.4, P(B)= p, P(AUB)=0.6 and And B are given to be independent events, find the value of P. 1/3
27) let A and B be two independent events. The probability of their simultaneous occurrence is 1/8 and the probability that neither occurs is 3/8. Find
A) P(A) .
B) P(B). 1/2, 1/4 or 1/4, 1/2
28) if A and B are two independent events such that P(A'∩B)=2/15 and P(A∩B')=1/6, then find
A) P(A)
B) P(B). 1/5, 1/6 or 5/6, 4/5
29) A dice is tossed twice. Find the probability of getting a number greater than 3 on each toss. 1/4
30) An unbiased die is tossed twice. Find the probability getting 4, 5, or 6 on the first toss and 1, 2, 3 or 4 on the second toss. 1/3
31) A die is thrown thrice. Find the probability of getting an odd number at least once. 7/8
32) two dice are thrown. Find the probability of getting an odd number on the first die and a the multiple of 3 on the other. 1/6
33) In two successive throws of a pair of dice, determine the probability of getting a total of 8 each time. 25/1296
34) A bag contains 3 red and 2 black balls. One ball is drawn from it at random. Its colour is noted and then it is put back in the bag. A second draw is made and the same procedure is repeated. Find the probability of drawing
A) two red balls. 9/25
B) two black balls. 4/25
C) first red and second block ball. 6/25
35) Two balls are drawn at random with replacement from a box containing 10 blacks and 8 red balls. Find the probability that
A) both 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
36) An urn contains 4 red and 7 black balls. Two balls are drawn at random with replacement. Find the probability of getting
A) two red balls. 16/121
B) two black balls. 49/121
C) 1 red and one black. 56/121
37) A bag contains 5 white.7 red and 4 Black balls. If 4 balls a drawn one by one with replacement, what is the probability that none is white ? (11/16)⁴
38) ) a bag contains 3 red and 5 black balls and a second bag contains 6 red and 4 Black balls. A ball is drawn from each bag. Find the probability that both are
A) red. 6/10
B) black. 9/40
39) A bag contains 5 white,7 red and 8 black balls. 4 balls are drawn one by one with replacement, what is the probability that at least one is white ? 1 -(3/4)⁴
40) Three cards are drawn with replacement from a well shuffled pack of cards. Find the probably that the cards drawn are king, queen and jack. 6/2197
41) ) A can hit a target 4 times in 5 shots, B 3 times in 4 shots, and C 2 times in 3 shots. calculate the probability that
A) A, B, C all may hit. 2/5
B) B, C may hit and A may not. 1/10
C) any two of A ,B and C will hit the target. 13/30
D) none of them will hit the target. 1/60
42) The probability that A hits a target is 1/3 and the probably that B hits it, is 2/5. What is the probability that the target will be hit, if each one of A and B shoots at the target ? 3/5
43) Given the probability that A can solve a problem is 2/3 and the probability that B can solve the same problem 3/5. Find the probability that none of the two will be able to solve the problem. 2/15
44) ) A problem in mathematics is given to 3 students whose chances of solving it are :1/2, 1/3, 1/4. What is the probability that problem is solved. 3/4
45) A can solve 90% of the problems given in a book and B can solve 70%. What is the probability that at least one of them will solve the problem, selected at random from the book ? 0.97
46) ) probabilities of solving a specific problem independently by A and B are 1/2 and 1/3 respectively. If both try to solve the problem independently, find the probability that.
A) the problem is solved. 2/3
B) exactly one of them solve the problem. 1/2
47) A and B are candidate seeking admission in a college. The probability that A is selected is 0.7 and the probability that exactly one of them is selected is 0.6. Find the probability that B is selected. 1/4
48) An article manufactured by a company consist of two parts X and Y. In the process of manufacture of the part X, 9 out of 100 parts may be defective. Similarly, 5 out of 100 are likely to be defective in the manufacture of part Y. Calculate the probability that the assembled product will not defective. 0.8645
49) An anti-craft gun can take a maximum of 4 shots at an enemy plane moving away from it. The probability of hitting the plane at the first, second, third and fourth shot are 0.4, 0.3, 0.2 and 0.1 respectively. What is the probability that gun hits the plane. 0.696
50) ) The probability of two students A and B coming to the school in time are 3/7 and 5/7 respectively. Assuming that the events, 'A coming in time and 'B coming in time are independent. Find the probability of only one of them coming to the school in time. Write at least one advantage of coming to school in time. 26/49
51) A class consists of 80 students, 25 of them are girls and 55 boys, 10 of them are rich and the remaining poor, 20 of them are fair complexioned. What is the probability of selecting a fair complexioned rich girl ? 5/512
52) policeman fires four bullets on a dacoit. The probability that the dacoit will be killed by any Bullet is 0.6. What is the probability that the dacoit is still alive ? 0.0256
53) The probably that a teacher will give an urn announced test during any class meeting is 1/5. if a student is absent twice, what is the probability that he will miss at least one test ? 9/25
54) A machine operates if all of its three components function. The probability that the first component fails during the year is 0.14, the second component fails is 0.10 and the third component fails is 0.05. What is the probability that the machine will fail during the year? 0.2647
55) A scientist has to make a decision on each of the two independent events I and II. Suppose the probability of error in a making decision on event I is 0.2 and that an event II is 0.05. Find the probability thst the scientist will make the correct decision on
A) both the events. 0.931
B) only one event. 0.068
56) A town has two fire extinguishing engines functioning independently. The probability availability of each engine, when needed is 0.95. What is the probability that
A) neither of them is available when needed? 0.0025
B) an engine is available when needed. 0.9975
C) exactly one engine is available when needed ? 0.095
57) A company has estimated that the probabilities of success for 3 products introduced in the market are 1/3, 2/5 and 2/3 respectively. Assuming independence, find the probability that
A) the three products are successful. 4/45
B) none of the products is successful. 2/15
58) The odds against A solving a certain problem are 4 to 3 and the odds in favour of B solving the same problem are 7 to 5. Find that the problem will be solved. 16/21
59) ) The odds against a certain event are 5 to 2 and the odds in favour of another event, independent to the former are 6 to 5. Find the probability that
A) atleast one of the events will occur. 52/77
B) none of the events will occur. 25/77
60) A coin is tossed and a dice is thrown. Find the probability that the outcome will be a head or a number greater than 4, or both. 2/3
Exercise - H
1) A bag contains 4 white and 2 black balls. Another contains 3 white and 5 black balls. If 1 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. 2 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) A bag contains 5 white and 3 black balls. 4 balls are successfully drawn out without replacement. What is the probability that they are alternatively of different colours ? 1/7
4) Bag A contains 4 red and 5 black balls and bag B contains 3 red and 7 black balls. One ball is drawn from bag A and two from bag B. Find the probability that out of 3 balls drawn, two are black and one is red. 7/15
5) A bag contains 5 red Marbles and 3 black Marbles. Three marbles are drawn one by one without replacement. What is the probability that atleast one of the three marbles drawn be black, if the first marble is red ? 25/56
6) A bag contains 3 white,3 black and 2 red balls. One by one, three balls are drawn without replaceing them. Find the probability the third ball is red. 1/4
7) There are three urns A,B and C. Urn A contains 4 white balls and 5 blue balls. Urn B contains 4 white balls and 3 blue balls. Urn C contains 2 white balls and 4 blue balls. One ball is drawn from each of these urns. What is the probability that out of these three balls drawn, two are white balls and one is a blue ball. 64/189
8) A bag contains 6 black and 3 white balls. Another bag contains 5 black and 4 white balls. If one ball is drawn from each bag, find the probability that these two balls are of the same colour. 14/27
9) A bag contain 3 red and 5 black balls and a second bag contains 6 red and 4 black balls. A ball is drawn from each bag. Find the probability that one is red and the other is black. 21/40
10) 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 the second is red. 20/81
C) one of them is black and other is red. 40/81
11) A bag contains 3 white, 4 red and 5 black balls. Two balls are drawn one after the other, without replacement. What is the probability that one is white and the other is black ? 5/22
12) A bag contains 8 red and 6 green balls. 3 balls are drawn one after another without replacement. Find the probability that atleast two positive drawn are green. 5/13
13) A bag conta8 7 white, 5 black and 4 red balls. Four balls are drawn without replacement. Find the probability that at least 3 balls are black. 23/364
14) A bag contains 4 white balls and 2 black balls. Another contains 3 white balls and 5 black balls. If one ball is drawn from each bag, find the probability that
A) both are white. 1/4, 5/24, 13/24
B) both are black.
C) one is white and one is black.
15) A bag contains 4 white, 7 black and 5 red balls. 4 balls are drawn with replacement. What is the probability that atleast two are white ? 67/256
16) A bag contains 4 red and 5 black balls, a second bag contains 3 red and 7 black balls. One ball is drawn at random from each bag, find the probability that the
A) balls are of different colours. 43/90
B) balls are of the same colour. 47/90
17) There are three urns A, B, C. Urn A contains 4 red balls and 3 black balls. Urn B contains 5 red balls and 4 black balls. Urn C contains 4 red and 4 black balls. 1 ball is drawn from each of these urns. What is the probability that 3 balls drawn consist of 2 red balls and a black balls? 17/42
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 at random with replacement. Find the probability of getting :
A) two red balls. 49/121
B) 2 Blue Balls. 16/121,56/121
C) one red and one blue ball.
20) There are 3 red and 5 black balls in bag 'A'; and 2 red and 3 black balls in bag 'B'. One ball is drawn from bag 'A' and two from bag 'B'. Find the probability that out of the 3 balls drawn one is red and 2 are black. 39/80
21) A card is drawn from a well shuffled deck of 52 cards. The outcome is noted, the card is replaced and the deck reshuffled. Another card is then drawn from the deck.
A) what is the probability that both the cards are of the same suit ? 1/4, 1/338
B) What is the probability that the first card is an ace and the second card is red Queen.
22) Two cards are drawn from a pack of 52 cards without replacement. What is the probability that one is red queen and the other is a king of black colour? 2/663
23) 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
24) cards are numbered 1 to 25. Two cards are drawn one after the other. Find the probability that the number of one card is multiple of 7 and on the other it is a multiple of 11. 1/50
25) Two cards and drawn successfully without replacement from a well shuffled deck of 52 cards. Find the probability of exactly one ace. 32/221
26) Two cards a 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
27) 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
28) 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 of the tickets is a multiple of 5 and on the other a multiple of 4. 4/45
29) The probability of student A passing an examination 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
30) The probability of a 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
31) The probability 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
32) Kamal and Monica appeared for an interview for two vacancies. The probability of Kamal's selection is 1/3 and that of Monica's selection 1/5. Find the probability that
A) both of them will be selected.
B) none of them will be selected.
C) at least one of them will be selected.
D) only one of them will be selected. 1/15,8/15,7/15,2/5
33) A certain team wins with probability 0.7, loses with probability 0.2 and ties with probability 0.1 the team plays 3 games. Find the probability that the team wins at least two of the games, but not lose. 0.49
34) 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
35) Arun and Tarun appeared for an interview for two vacancies. The probability of Arun's selection is 1/4 and that of Tarun's rejection is 2/3. Find the probability that atleast one of them will be selected. 1/2
36) 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 the wife's selection is 1/5. What is the probability that:
A) both of them will be selected?
B) only one of them will be selected ? 1/35,2/7,24/35
C) none of them will be selected ?
37) A,B and C shrt to hit a target. If A hit the target 4 times and 5 trials; B hits it three times in four trials and C hits 2 times in three trials; what is the probability that the target is hit by at least two persons? 5/6
38) Three groups of children contains three girls and one boy; two girls and two 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
39) Thee 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
40) A speaks truth in 60% of the cases and B in 90% of the cases. In what percentage of cases are they likely
A) contradict each other in stating same fact ? 42%
B) agree in stating the same fact ? 58%
41) A speaks 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%
42) 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 a alive 25 years hence,. 35/
D) atleast one of them will be alive 25 years hence. 61/96
43) A clerk was asked to mail 3 report cards to 3 students. He addresses 3 envelopes but unfortunately paid no attention to which report card be put in which envelope. what is the probability that exactly one of the student received his her own card ? 1/2
44) Neelam 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
45) A doctor claims that 60% of the patient 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. 216/625
B) none of his next 4 patients is allergic to weeds ? 16/625
46) Two persons A and B throw a dice alternatively till one of them gets a '3' and wins the game. Find their respectively Probabilities of winning, if A begins. 5/11
47) A and B throw alternatively a pair of dice. A wins if he throws 6 before B throws 7 and B wins if the throw 7 before A throws 6. Find their respective chance of winning, if A begins. 31/61
48) 3 person A,B,C throw a dice in succession will one gets a '6' and wins the game. Find their respective Probabilities of winning, if a begins. 25/91
49) A and B toss a Coin alternatively till one of them gets a head and wins the game find the probability that we will win the game. If A starts the game, find the probability that B will win the game. 1/3
50) 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 same fact. 0.44
51) A,B, C are independent witness of an event which is a 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 witnesses? 107/120
52) 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 all subjects. 0.03
B) grade A in no subject. 0.28
C) grade A in two subjects. 0.22
53) A and B tame turns in throwing two dies, the first to throw 9 being awarded the prize. Show that their chance of winning are in the ratio 9:8.
54) 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 indefinitely. 4/7,2/7,1/7
55) 3 person A,B,C throw a dice in succession till one gets a '6' and wins the game. Find their respective probabilities of winning. 36/91,30/91,25/91
56) 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.
57) Fatima and John appear in an interview for two vacancies for the same post. The probability of Fatima's selection 1/7 and that of John's selection is 1/5. What is the probability that:
A) both of them will be selected.
B) only one of them will be selected ? 1/35,2/7,24/35
C) none of them will be selected ?
58) 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 ?
B) you both enter the different sections? 17/33, 16/33
59) In a hockey match, both teams A and B scored same number of goals upto the end of the game, so to decide the winner, the referee asked both the captains to throw a die alternatively and decide that the team, whose captain gets a first six, will be declared the winner. If the captain of Team A was asked to start, find their respective Probabilities of winning the match and state whether the decision of the refree was fair or not.
Team A:6/11, Team B: 5/11; The decision was fair as the two Probabilities are almost equal.
Exercise - I
Continue....
BAYE'S THEOREM
Exercise -J
1) In a bolt factory, machines A,B,C manufacture 25%,35%,40% of the total bolts. Of their output 5,4 and 2% are 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 it is manufactured by the machine B? 28/69
2) Three Urns contains 6 red 4 black; 4 red, 6 black, and 5 red and 5 black balls. 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 is drawn from the first urn. 2/5
3) There are 3 bags, each containing 5 white and 3 black balls. Also there are 2 bags, each containing 2 white 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
4) Urn-1 contains 5 red and 5 black balls, Urn-2 contains 4 red and 8 black balls and Urn-3 contains 3 red and 6 black balls. One urn is chosen at random and a ball is drawn. The colour of the ball is black. What is the probability that is has been drawn from Urn-3? 4/11
5) 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
6) In a test, an examinee either guesses or copies or knows the answer to a multiple choice question with four 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
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) An insurance company insured 6000 scooter- drivers, 3000 car drivers and 9000 truck drivers. The probability of an accident involving a scooter, a car and a truck is 0.02, 0.06 and 0.03 respectively. One of the insured persons meets with an accident. Find the probability that he is a car driver. 6/19
9) A firm produces steel pipes in three plants A, B and C, with daily production of 500, 1000 and 2000 units respectively. If is known that fractions of defective output produced by three are respectively .005, .008, and.010. A pipe is selected at random from a day's total production and found to be defective. What is the probability that it came from the first plant. 5/61
10) Marksmen A and B compete by taking turns to shoot at a target. Odds in favour of A hitting the target (in a single try) are 3:2 and the odds in favour of B hitting the target (in a single try) are 4:3.
Calculate the probability of A winning the competition if he gets the first chance to shoot. 21/29
11) A letter is known to have come either from TATANAGAR or CALCUTTA. On the envelope 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
EXERCISE - K
MATHEMATICAL EXPECTATION
1) a) A random variable x has the following distribution, find the expectation, Mean, S. D Variance.
x: 0 1 2 3
P: 1/8 3/8 3/8 1/8. 1.5 0.87
b) x: 2 3 4 5
P: 0.2 0.4 0.3 0.1 3.3,0.81
c) x: 4 6 7 10
P: 0.2 0.4 0.3 0.1 6.3, 1.62
2) A card is drawn at random from a pack of 52 cards. if ace counts one, king, queen and jack count 10 each and other count at their face value, find the expectation of the value of the card. 85/13
3) Find the expected number of tails in tossing three coins. 1.5
4) Find the mathematical expectation of number of tails in thowing two coins. 1
5) 4 coins are tossed. If x denotes the number of heads, find the expected value and variance of x. 2,1
6) If a coin is tossed 100 times, in how many times will you expected head ? 50
7) A boy throws a coin four times and guesses each time whether the head or the tail has been thrown. He was not allowed to see the result. He is to receive ₹2 for 2 heads, ₹4 for 3 heads and ₹6 for 4 heads. Find his expectation. 2.13
8) A man draws two balls from a bag containing 3 white and 5 black balls. If he is to receive ₹14 for every white balls and ₹7 for every black balls, what is his expectation ? 19.25
9) A bag contains 5 white and 7 black balls. Find the expectation of a man who is allowed to draw two balls from the bag and who is receive Rs 1 for each black ball and ₹2 for each white ball drawn. 2.83
10) A box contains 8 tickets. 3 of the tickets carry a prize of ₹5 each the other 5 a prize of ₹2 each.
A) If one ticket is drawn what is the expected value of the prize ?
B) if two tickets are drawn what is the expected value of the game ? ₹3.12, ₹6.25
11) A box contains 4 white and 6 red balls. If 2 balls are drawn from it, then find the mathematical expectation of the number of red balls. 1.2
12) A bag contains 2 white balls and 3 black balls. Four persons A, B, C and D in this order, draw a ball from the bag and do not replace it. The first person to draw a white ball is to receive ₹20. Determine their expectation. ₹8,6, 4, 2
13) A box contains 4 red and 3 white balls. 2 balls are drawn from it and a man is to receive ₹20 for each white ball and lose ₹10 for each red ball. Find the amount expected by the man from the game. ₹2.86
14) In a box there are 5 watches of which 2 watches are known to be defective. 2 watches are taken out at random. Let X denote the number of defective watches selected. Obtain the probability distribution of X. Also calculate E(X). 4/5
15) Find the mathematical expectation of the number of points if a balance die is rolled. 3.5
16) The probability that there is atleast one error in an account statement prepared by A is 0.3 and for B and C, they are 0.4 and 0.45 respectively. A, B and C prepared 20, 10, 40 statements respectively. Find the expected number of correct statements in all. 42
17) Find the expected value and variance of number of points in rolling two balanced dice. 7, 5.833
18) Find the expected value of the product of the points on two dice. 49/4
19) A number is chosen at random from the set 1,2,3,...100 and another number is chosen at random from the set 1, 2,3,....50. what is the expected value of the product? 1287.75
20) A and B toss in turn an ordinary die for a prize of ₹55. The first to toss a six wins. If A has the first throw what is his expectation. ₹30
21) A and B play for a prize of ₹99. The prize is to be own by a player who first throw a '3' with one die. A first throw and if he fails B throws and if B fails A again throws, and so on. Find their respective expectations. 54, 45
22) The monthly demand for TV sets is known to have the following probability distribution:
Demand (x): 1 2 3 4 5
Prob(p): 0.10 0.15 0.20 0.35 0.20 The total cost (₹ y) of producing x TV sets is given by y= 100000 + 2000x. Find expected demand and expected cost. 3.40, 106800
23) A and B play for a prize of ₹152. They throw a pair of dice in succession. A win if he throws 8 before B throws 9 and B wins if he throws A throws 8. If A begins, find their respective expectations. 90, 62
Multiple choice questions
1) One card dron drawn from a pack of 52 cards. The probability that it is the card of a king or spade is
A) 1/26 B) 3/26 C) 4/13 D) 3/13
2) Two dice are thrown together. The probability that atleast one will show its digit greater than 3 is
A) 1/4 B) 3/4 C) 1/2 D) 1/8
3) Two dice are thrown together. The probability of getting a total score of 5 is
A) 1/18 B) 1/12 C) 1/9 D) none
4) Two dice are thrown together. The probability of getting total score of seven is
A) 5/36 B) 6/36 C) 7/36 D) 8/36
5) The probability of getting a total of 10 in a single throw of two dice is
A) 1/9 B) 1/12 C) 1/6 D) 5/36
6) A card is drawn at random from a pack of 100 cards numbered 1 to 100. The probability of drawing a number which is a square is
A) 1/5 B) 2/5 C) 1/10 D) none
7) A bag contains 3 red, 4 white and 5 blue balls. All balls are different. Two balls are drawn at random. The probability that they are of different colour is
A) 46/66 B) 10/33 C) 1/3 D) 1
8) Two dice are thrown together. The probability that neither they show equal digits nor the sum of their digits is 9 will be
A) 13/15 B) 13/18 C) 1/9 D) 8/9
9) Four persons are selected at random out of 3 men, 2 women and 4 children. The probability that there are exactly 2 children in the selection is
A) 11/21 B) 9/21 C) 10/21 D) none
10) The probabilities of happening of two events A and B are 0.25 and 0.50 respectively. If the probability of happening of A and B together is 0.14, then Probability that neither A nor B happens is
A) 0.39 B) 0.25 C) 0.11 D) none
11) A die is thrown, then the probability that a number 1 or 6 may occur is
A) 2/3 B) 5/6 C) 1/3 D) 1/2
12) Six boys and six girls sit in a row randomly. The probability that all girls sit together is
A) 1/122 B) 1/112 C) 1/102 D) 1/132
13) The probabilities of three mutually exclusive events A, B and C are given by 2/3, 1/4 and 1/6 respectively. The statement
A) is true B) is false C) nothing can be said D) could be either
14) (1-3p)/2, (1+ 4p)/3, (1+ p)/6 are the probabilities of three mutually exclusive and exhaustive events, then the set of all values of p is
A) (0,1) B) (-1/4,1/3) C) (0,1/3) D) n
15) A pack of cards contains 4 aces, 4 kings, 4 queens and 4 jacks. Two cards are drawn at random. The probability that atleast one of them is an ace is
A) 1/5 B) 3/16 C) 9/20 D) 1/9
16) If three dice are thrown simultaneously, then the probability of getting a score of 5 is
A) 5/216 B) 1/6 C) 1/36 D) none
17) One of the two events must occur. If the chance of one is 2/3 of the other, then odd in favour of the other are
A) 1:3 B) 3:1 C) 2:3 D) 3:2
18) The probability that a leap year will have 53 Friday or Saturday is
A) 2/7 B) 3/7 C) 4/7 D) 1/7
19) A person write 4 letters and addresses 4 envelopes. If the letter are placed in the envelopes at random, then the probability that all letters are not placed in the right envelopes, is
A) 1/4 B) 11/24 C) 15/24 D) 23/24
20) A and B are two events such that P(A)= 0.25 and P(B)= 0.50. the probability both happening together is 0.14. the probability of both A and B not happening is
A) 0.39 B) 0.25 C) 0.11 D) none
21) If the probability of A to fail in an examination is 1/5 And that of B is 3/10. Then, the probability that either A or B fails is
A) 1/2 B) 11/25 C) 19/50 D) none
22) A box contains 10 good article and 6 defective articles. One item is drawn at random. The probability that it is either good or has a defect, is
A) 64/64 B) 49/64 C) 40/64 D) 24/64
23) Three integers are selected at random from the first 20 integers. The probability that their product is even is
A) 2/19 B) 3/29 C) 17/19 D) 4/19
24) Out of 30 consecutive integers, 2 are chosen at random. The probability that their sum is odd, is
A) 14/29 B) 16/29 C) 15/29 D) 10/29
25) A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is selected randomwise, the probability that it is black or red ball is
A) 1/3 B) 1/4 C) 5/12 D) 2/3
26) Two dice are thrown simultaneously. The probability of getting a pair of aces is
A) 1/36 B) 1/3 C) 1/6 D) none
27) An urn contains 9 balls two of which are red, three blue and four black. Three balls are drawn at random. The probability that they are of the same colour is
A) 5/84 B) 3/9 C) 3/7 D) 7/17
28) Five 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 floor is
A) 7P5/7⁵ B) 7⁵/7P5 C) 6/6P5 D) 5P5/5⁵
29) A box contains 10 good article and 6 defective. One item is drawn at random. The probability that it is either good or has a defect is
A) 64/64 B) 49/64 C) 40/64 D) 24/64
30) A box contains 6 bolts and 10 nuts. Half of the bolts and half of the nuts are rusted. If one item is chosen at random, the probability that it is rusted or is a bolt is
A) 3/16 B) 5/16 C) 11/16 D) 14/16
31) If S is the sample space and P(A)= 1/3 P(B) and S= A U B, where A and B are two mutually exclusive events, then P(A) is
A) 1/4 B) 1/2 C) 3/4 D) 3/8
32) Of P(A UB)= 0.8 and P(A intersection B)= 0.3, then P(A')+ P(B') is
A) 0.3 B) 0.5 C) 0.7 D) 0.9
