# RD Sharma Solution CLass 12 Mathematics Chapter 31 Probability

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

## Chapter 31 – Probability Exercise Ex. 31.2

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

An urn contains 10 black and 5 white balls. Two balls are drawn from the urn one after the other without replacement. What is probability that both drawn balls are black?

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

## Chapter 31 – Probability Exercise Ex. 31.3

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

If A and B are two events such that

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Assume that 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

(i) the youngest is a girl

(ii) at least one is girl.

Solution 30

(i) Let ‘A’ be the event that both the children born are girls.

Let ‘B’ be the event that the youngest is a girl.

We have to find conditional probability P(A/B).

(ii) Let ‘A’ be the event that both the children born are girls.

Let ‘B’ be the event that at least one is a girl.

We have to find the conditional probability P(A/B).

## Chapter 31 – Probability Exercise Ex. 31.4

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Given that the events ‘A coming in time’ and ‘B coming in time’ are independent.

The advantage of coming to school in time is that you will not miss any part of the lecture and will be able to learn more.

Question 28

Two dice are thrown together and the total score is
noted. The event 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 pairs of events, if any, are independent.

Solution 28

Question 29

Let A and B be two independent events such that P (A) =
p1 and P (B) = p2. Describe in words the events whose
probabilities are:

(i) p1p2
(ii) (1 – p1)p2 (iii) 1-(1- p1) (1 – p2)
(iv) p1 + p2 = 2p1p2

Solution 29

## Chapter 31 – Probability Exercise Ex. 31.5

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

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 refree asked both the captains to throw a die alternately 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.

Solution 35

## Chapter 31 – Probability Exercise Ex. 31.6

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

There machines E1, E2, E3
in a certain factory produce 50%, 25% and 25% respectively, of the total
daily output of electric bulbs. It is known that 4% of the tubes produced one
each of machines E1 and E2 are defective, and that 5%
of those produced on E3 are defective. If one tube is picked up at
random from a day’s production, calculate the probability that it is
defective.

Solution 13

## Chapter 31 – Probability Exercise Ex. 31.7

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Suppose a girl throws a die. If she gets 1 or 2, she
tosses a coin three times and notes the number of tails. If she gets 3,4, 5
or 6, she tosses a coin once and notes whether a ‘head’ or ‘tail’ is
obtained. If she obtained exactly one ‘tail’, what is the probability that
she threw 3, 4, 5 or 6 with the die?

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

An item is manufactured by three machine A, B and C.
out of the total number of items manufactured during a specified period, 50%
are manufacture on machine A 30% on B and 20% on C. 2% of the items produced
on A and 2% of items produced on B are defective and 3% of these produced on
C are defective. All the items stored at one godown.
One items is drawn at random and is found to be
defective. What is the probability that it was manufactured on machine A?

Solution 14

Question 15

There are three coins. One is two-headed coin (having head on both faces), another is biased coin that comes up heads 75% of the times and third is also a biased coin that comes up tail 40% of the times. One of the three coins is chosen at random and tossed, and it shows heads. What is the probability that it was the two-headed coin?

Solution 15

Question 16

Solution 16

Question 17

Solution 17

Question 18

Solution 18

Question 19

In a group of 400 people, 160 are smokers and non-vegetarian, 100 are smokers and vegetarian and the remaining are non-smokers and vegetarian. The probabilities of getting a special chest disease are 35%, 20% and 10% respectively. A person is chosen from the group at random and is found to be suffering from the disease. What is the probability that the selected person is a smoker and non-vegetarian?

Solution 19

Question 20

Solution 20

Question 21

Solution 21

Question 22

Solution 22

Question 23

Solution 23

Question 24

Solution 24

Question 25

Solution 25

Question 26

Solution 26

Question 27

Solution 27

Question 28

Solution 28

Question 29

Solution 29

Question 30

Solution 30

Question 31

Solution 31

Question 32

Solution 32

Question 33

Solution 33

Question 34

Solution 34

Question 35

Solution 35

Question 36

Solution 36

Question 1

Solution 1

Question 2

Solution 2

Question 3

Solution 3

Question 4

Solution 4

Question 5

Solution 5

Question 6

Solution 6

Question 7

Solution 7

Question 8

Solution 8

Question 9

Solution 9

Question 10

Solution 10

Question 11

Solution 11

Question 12

Solution 12

Question 13

Solution 13

Question 14

Solution 14

Question 15

Solution 15

## Chapter 31 – Probability Exercise MCQ

Question 1

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

Solution 1

Correct option: (a)

Question 2

A and B draw two cards each, one after another, from a pack of well-shuffled pack of 52 cards. The probability that all the four cards drawn are of the same suit is

Solution 2

Correct option: (a)

Question 3

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

a. 0.39

b. 0.25

c. 0.11

d. none of these

Solution 3

Correct option: (a)

Question 4

Solution 4

Correct option: (b)

Question 5

Indian play two matches each with West Indies and Australia. In any match the probabilities of India getting 0,1and 2 points are 0.45, 0.05 and 0.50 respectively. Assuming that the outcomes are independent, the probability of India getting at least 7 points is

a. 0.0875

b. 1/16

c. 0.1125

d. None of these

Solution 5

Correct option: (a)

Question 6

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

Solution 6

Correct option: (a)

Question 7

The probability that a leap year will have 53 Friday or 53 Saturday is

Solution 7

Correct option: (b)

Question 8

A person writes 4 letters and addresses 4 envelopes. If the letters are placed in the envelopes at random, then the probability that all letters are not placed in the right envelopes, is

Solution 8

Correct option: (d)

Question 9

A speaks truth in 75% cases and B speaks truth in 80% cases. Probability that they contradict each other in a statement, is

Solution 9

Correct option: (a)

Question 10

Three integers are chosen at random from the first 20 integers. The probability that their product is even is

Solution 10

Correct option: (c)

Question 11

Out of 30 consecutive integers, 2 are chosen at random. The probability that their sum is odd, is

Solution 11

Correct option: (c)

Question 12

A bag contains 5 black balls, 4 white balls and 3 red balls. If a ball is selected random wise, the probability that it is black or red ball is

Solution 12

Correct option: (d)

Question 13

Two dice are thrown simultaneously. The probability of getting a pair of aces is

Solution 13

Correct option: (a)

Question 14

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

Solution 14

Correct option: (a)

Question 15

A coin is tossed three times. If events A and B defined as A = Two heads come, B = Last should be head. Then , A and B are

a. independent

b. dependent

c. both

d. mutually exclusive

Solution 15

Correct option: (b)

Question 16

Five persons entered the life 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 floors beginning with the first, then probability of all 5 persons leaving different floor is

Solution 16

Correct option: (a)

Question 17

A box contains 10 goods articles and 6 with defects. One item is drawn at random. The probability that it is either or has a defect is

Solution 17

Correct option: (a)

Question 18

A box contains 6 nails and 10 nuts. Half of the nails and half of the nuts are rusted. If one item is chosen at random, the probability that it is rusted or is a nail is

Solution 18

Correct option: (c)

Question 19

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

Solution 19

Correct option: (d)

Question 20

a. 1/4

b. 1/2

c. 3/4

d. 3/8

Solution 20

Correct option: (a)

Question 21

Solution 21

Correct option: (d)

Question 22

a. 0.3

b. 0.5

c. 0.7

d. 0.9

Solution 22

Correct option: (d)

Question 23

A bag X contains 2 white and 3 black balls and another bag Y contains 4 white and 2 black balls. One bag is selected at random and a ball is drawn from it. Then, the probability chosen to be white is

a. 2/15

b. 7/15

c. 8/15

d. 14/15

Solution 23

Correct option: (c)

Question 24

Two persons A and B turn in throwing a pair of dice. The first person to throw 9 from both dice will be awarded the prize. If A throws first, then the probability that B wins the game is

a. 9/17

b. 8/17

c. 8/9

d. 1/9

Solution 24

Correct option: (b)

Question 25

The probability that in a year of 22nd century chosen at random, there will be 53 Sundays, is

a. 3/28

b. 2/28

c. 7/28

d. 5/28

Solution 25

Question 26

From a set of 100 cards numbered 1 to 100, one card is draw at random. The probability that the number obtained on the card is divisible by 6 or 8 but not by 24 is

a. 6/25

b. 1/4

c. 1/6

d. 2/5

Solution 26

Correct option: (a)

Question 27

a. 1/10

b. 1/8

c. 7/8

d. 17/20

Solution 27

Correct option: (c)

Question 28

a. 14/17

b. 17/20

c. 7/8

d. 1/8

Solution 28

Correct option: (a)

Question 29

Associated to a random experiment two events A and B are such that P (B) = 3/5, P (A/B) = 1/2 and (A B) = 4/5. The Value of P(A) is

Solution 29

Correct option: (b)

Question 30

If P(A) = 3/10, P(B) = 2/5 and P(A B) = 3/5, then P(A/B) + P(B/A) equals

a. 1/4

b. 7/2

c. 5/12

d. 1/3

Solution 30

Correct option: (b)

Note: option is modified.

Question 31

a. 5/9

b. 4/9

c. 4/13

d. 6/13

Solution 31

Correct option: (a)

Question 32

Solution 32

Correct option: (c)

Question 33

Solution 33

Correct option: (c)

Question 34

Solution 34

Correct option: (d)

Question 35

Solution 35

Correct option: (d)

Question 36

If P(A) = 0.4, P(B) = 0.8 and P(B/A)=0.6, then

P(A B)=

a. 0.24

b. 0.3

c. 0.48

d. 0.96

Solution 36

Correct option: (d)

Question 37

Solution 37

Correct option: (d)

Question 38

Solution 38

Correct option: (d)

Question 39

If A and B are two events such that A ϕ, B = ϕ, then

Solution 39

Correct option: (a)

Question 40

Solution 40

Correct option: (c)

Question 41

If the events A and B are independent, then P(A B) is equal to

a. P(A) + P(B)

b. P(A) – P(B)

c. P(A) P(B)

d.

Solution 41

Correct option: (c)

P(A B)= P(A) P(B) for independent events.

Question 42

Solution 42

Correct option: (d)

Question 43

If A and B are two independent events such that P(A) = 0.3, P(A B) = 0.5, then P(A/B) – P(B/A) =

Solution 43

Correct option: (c)

Question 44

A flash light has 8 batteries out of which 3 are dead. If two batteries are selected without replacement and tested, the probability that both are dead is

Solution 44

Correct option: (a)

Question 45

A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement, then the probability of getting exactly one red ball is

Solution 45

Correct option: (b)

Question 46

A bag contains 5 red and 3 blue balls. If 3 balls are drawn at random without replacement, then the probability that exactly two of the three balls were red, the first ball being red, is

Solution 46

Correct option: (b)

Question 47

In a college 30% students fail in physics, 25% fails in Mathematics and 10% fail in both. One student is chosen at random. The probability that she fails in Physics if she has failed in Mathematics in

Solution 47

Correct option: (c)

Question 48

Three persons A, B and C fire a target in turn starting with A. their probabilities of hitting the target are 0.4, 0.3 and 0.2 respectively. The probability of two hits is

a. 0.024

b. 0.452

c. 0.336

d. 0.138

Solution 48

Question 49

Solution 49

Correct option: (a)

Question 50

Two cards are drawn from a well shuffled deck of 52 playing cards with replacement. The probability that both cards are queen is

Solution 50

Correct option: (a)

Question 51

A box contains 3 orange balls, 3 green balls and 2 blue balls. Three balls are drawn at random from the box without replacement. The probability of drawing 2 green balls and one blue ball is

Solution 51

Correct option: (d)

Question 52

If two events are independent, then

a. they must be mutually exclusive

b. the sum of their probabilities must be equal to 1

c. (a) and (b) both are correct

d. none of the above is correct

Solution 52

Correct option: (d)

Question 53

Two dice are thrown. If it is known that the sum of the numbers on the dice was less than 6, the probability of getting a sum 3, is

Solution 53

Correct option: (c)

Question 54

Solution 54

Correct option: (b)

Question 55

Solution 55

Correct option: (c)

Question 56

A die is thrown and a card is selected at random from a deck of 52 playing cards. The probability of getting an even number of the die and a spade card is

Solution 56

Correct option: (c)

Question 57

Assume that in a family, each child is equally likely to be a boy or a girl. A family with three children is chosen at random. The probability that the eldest child is a girl given that the family has at least one girl is

Solution 57

Correct option: (d)

Question 58

Let A and B be two events. If P(A) = 0.2, P(B) = 0.4,

P(A B) = 0.6, then P(A/B) is equal to

a. 0.8

b. 0.5

c. 0.3

d. 0

Solution 58

Correct option: (d)

Question 59

Solution 59

Correct option: (c)