# RD SHARMA Solutions for Class 12-science Maths Chapter 31 - Probability

## Chapter 31 - Probability Exercise Ex. 31.1

## Chapter 31 - Probability Exercise Ex. 31.2

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?

## Chapter 31 - Probability Exercise Ex. 31.3

If A and B are two events such that

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.

(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

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.

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.

Let A and B be two independent events such that P (A) =
p_{1} and P (B) = p_{2}. Describe in words the events whose
probabilities are:

(i) p_{1}p_{2}
(ii) (1 - p_{1})p_{2} (iii) 1-(1- p_{1}) (1 - p_{2})
(iv) p_{1} + p_{2} = 2p_{1}p_{2}

## Chapter 31 - Probability Exercise Ex. 31.5

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.

## Chapter 31 - Probability Exercise Ex. 31.6

There machines E_{1}, E_{2}, E_{3}
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 E_{1} and E_{2} are defective, and that 5%
of those produced on E_{3} are defective. If one tube is picked up at
random from a day's production, calculate the probability that it is
defective.

## Chapter 31 - Probability Exercise Ex. 31.7

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?

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?

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?

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?

## Chapter 31 - Probability Exercise Ex. 31VSAQ

## Chapter 31 - Probability Exercise MCQ

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

Correct option: (a)

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

Correct option: (a)

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

Correct option: (a)

Correct option: (b)

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

Correct option: (a)

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

Correct option: (a)

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

Correct option: (b)

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

Correct option: (d)

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

Correct option: (a)

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

Correct option: (c)

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

Correct option: (c)

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

Correct option: (d)

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

Correct option: (a)

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

Correct option: (a)

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

Correct option: (b)

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

Correct option: (a)

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

Correct option: (a)

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

Correct option: (c)

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

Correct option: (d)

a. 1/4

b. 1/2

c. 3/4

d. 3/8

Correct option: (a)

Correct option: (d)

a. 0.3

b. 0.5

c. 0.7

d. 0.9

Correct option: (d)

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

Correct option: (c)

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

Correct option: (b)

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

NOTE: Answer not matching with back answer.

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

Correct option: (a)

a. 1/10

b. 1/8

c. 7/8

d. 17/20

Correct option: (c)

a. 14/17

b. 17/20

c. 7/8

d. 1/8

Correct option: (a)

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

Correct option: (b)

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

Correct option: (b)

Note: option is modified.

a. 5/9

b. 4/9

c. 4/13

d. 6/13

Correct option: (a)

Correct option: (c)

Correct option: (c)

Correct option: (d)

Correct option: (d)

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

Correct option: (d)

Correct option: (d)

Correct option: (d)

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

Correct option: (a)

Correct option: (c)

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.

Correct option: (c)

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

Correct option: (d)

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) =

Correct option: (c)

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

Correct option: (a)

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

Correct option: (b)

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

Correct option: (b)

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

Correct option: (c)

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

NOTE: Answer not matching with back answer.

Correct option: (a)

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

Correct option: (a)

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

Correct option: (d)

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

Correct option: (d)

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

Correct option: (c)

Correct option: (b)

** **

Correct option: (c)

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

Correct option: (c)

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

Correct option: (d)

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

Correct option: (d)

Correct option: (c)

NOTE: Answer not matching with back answer.

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