NCERT Solutions For Class 12 Maths Chapter 13 Probability

NCERT Solutions for Class 12 Maths Chapter 13 Probability is designed and prepared by the best teachers across India. All the important topics are covered in the exercises and each answer comes with a detailed explanation to help students understand concepts better. These NCERT solutions play a crucial role in your preparation for all exams conducted by the CBSE, including the JEE.

Chapter 13 Probability Solutions covers multiple exercises. The answer to each question in every exercise is provided along with complete, step-wise solutions for your better understanding. This will prove to be most helpful to you in your home assignments as well as practice sessions.
The topics and sub-topics included in Chapter 13 Probability the following:

Section Name	Topic Name
13	Probability
13.1	Introduction
13.2	Conditional Probability
13.3	Multiplication Theorem on Probability
13.4	Independent Events
13.5	Bayes’ Theorem
13.6	Random Variables and its Probability Distributions
13.7	Bernoulli Trials and Binomial Distribution

Table of Contents

NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.1

Question 1
Given that E and Fare events such that
P (E) = 0.6, P (F) = 0.3 and P(E∩F) = 0.2
find P(E|F) and P (F|E).
Solution:

Question 2
Compute P(A|B) if P(B)=0.5 and P (A∩B) = 0.32.
Solution:

Question 3.
If P (A)=0.8, P (B)=0.5 and P(B/A)=0.4, find
(i) P(A∩B)
(ii) P(A/B)
(iii)P(A∪B)
Solution:

Question 4.
Evaluate P(A∪B) if 2P(A) = P(B) = $\frac { 5 }{ 13 }$ and P(A|B) = $\frac { 2 }{ 5 }$ .
Solution:

Question 5.
If P(A) = $\frac { 6 }{ 11 }$ ,P(B) = $\frac { 5 }{ 11 }$ and P(A∪B) = $\frac { 7 }{ 11 }$ ,
Find
(i) P(A∩B)
(ii) P(A|B)
(iii) P(B|A)
Solution:

Question 6.
Determine P(E/F) in question 6 to 9:
A coin is tossed three times, where
(i) E: head on third toss F: heads on first two tosses.
(ii) E: at least two heads F : at most two heads
(iii) E: at most two tails F: at least one tail
Solution:

Question 7.
Two coins are tossed once
(i) E: tail appears on one coin F: one coin shows head
(ii) E: no tail appears F: no head appears.
Solution:

Question 8.
A die is thrown three times.
E: 4 appears on the third toss
F: 6 and 5 appears respectively on first two tosses.
Solution:

Question 9.
Mother, father and son line up at random for a family picture:
E: son on one end, F: father in middle
Solution:

Question 10.
A Mack and a red die are rolled.
(a) Find the conditional probability of obtaining a sum greater than 9, given that the black die resulted in a 5.
(b) Find the conditional probability of obtaining the sum 8, given that the red die resulted in a number less than 4.
Solution:

Question 11
A fair die is rolled. Consider events E = {1,3,5} F = {2,3} and G = {2,3,4,5}, Find
(i) P(E|F) and P(F|E)
(ii) P(E|G) and P(G|E)
(iii) P((E ∪ F) |G) and P (E ∩ F)|G)
Solution:

Question 12.
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
(i) the youngest is a girl,
(ii) at least one is girl?
Solution:

Question 13.
An instructor has a question bank consisting of300 easy True/ False questions, 200 difficult True/ False questions, 500 easy multiple choice questions and 400 difficult multiple choice questions. If a question is selected at random from the question bank, what is the probability that it will be an easy question given that it is a multiple choice question?
Solution:

Question 14.
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’.
Solution:

Question 15.
Consider the experiment of throwing a die, if a multiple of 3 comes up, throw the die again and if any other number comes, toss a coin. Find the conditional probability of the event ‘the coin shows a tail’ given that ‘at least one die shows a 3’.
Solution:

In each of the following choose the correct answer:

Question 16.
If P(A) = $\frac { 1 }{ 2 }$ , P (B) = 0 then P (A | B) is
(a) 0
(b) $\frac { 1 }{ 2 }$
(c) not defined
(d) 1
Solution:

Question 17.
If A and B are events such that P(A | B) = P(B | A) then
(a) A⊂B but A≠B
(b) A = B
(c) A∩B = φ
(d) P(A) = P(B)
Solution:

NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.2

Question 1
If, P(A) =3/5 and P(B) =1/5 find P (A ∩ B) if A and B are independent events.
Solution:

Question 2.
Two cards are drawn at random and without replacement from a pack of 52 playing cards.Find the probability that both the cards are black.
Solution:

Question 3.
A box of oranges is inspected by examining three randomly selected oranges drawn without replacement. If all the three oranges are good, the box is approved for sale otherwise it is rejected. Find the probability that a box containing 15 oranges out of which 12 are good and 3 are bad ones will be approved for sale.
Solution:

Question 4.
A fair coin and an unbiased die are tossed. Let A be the event ‘head appears on the coin’ and B be the event ‘3 on the die’. Check whether A and B are independent events or not
Solution:

Question 5.
A die marked 1,2,3 in red and 4,5,6 in green is tossed. Let A be the event, ‘the number is even’, and B be the event, ‘the number is red’. Are A and B independent?
Solution:

Question 6.
Let E and F be the events with P(E) = $\frac { 3 }{ 5 }$ , P (F) = $\frac { 3 }{ 10 }$ and P (E ∩ F) = $\frac { 1 }{ 5 }$ . Are E and F independent?
Solution:

Question 7.
Given that the events A and B are such that P(A) = $\frac { 1 }{ 2 }$ ,P(A∪B) = $\frac { 3 }{ 5 }$ and P(B) = p. Find p if they are
(i) mutually exclusive
(ii) independent.
Sol:

Question 8.
Let A and B independent events P(A) = 0.3 and P(B) = 0.4. Find
(i) P(A∩B)
(ii) P(A∪B)
(iii) P (A | B)
(iv) P(B | A)
Solution:

Question 9.
If A and B are two events, such that P (A) = $\frac { 1 }{ 4 }$ , P(B) = $\frac { 1 }{ 2 }$ ,and P(A∩B) = $\frac { 1 }{ 8 }$ .Find P (not A and not B)
Solution:

Question 10
Events A and B are such that
P(A) = $\frac { 1 }{ 2 }$ ,P(B) = $\frac { 7 }{ 12 }$ and P (not A or not B) = $\frac { 1 }{ 4 }$ . State whether A and Bare independent
Solution:

Question 11
Given two independent events A and B such that P (A) = 0.3, P(B) = 0.6. Find
(i) P(A and B)
(ii) P(A and not B)
(iii) P (A or B)
(iv) P (neither A nor B)
Solution:

Question 12
A die is tossed thrice. Find the probability of getting an odd number at least once.
Solution:

Question 13
Two balls are drawn at random with replacement from a box containing 10 black and 8 red balls. Find the probability that
(i) both balls are red.
(ii) the first ball is “black and the second is red.
(iii) one of them is black and other is red.
Solution:

Question 14
Probability of solving specific problem independently by A and B are $\frac { 1 }{ 2 }$ and $\frac { 1 }{ 3 }$ respectively. If both try to solve the problem independently, find the probability that
(i) the problem is solved
(ii) exactly one of them solves the problem.
Solution:

Question 15
One card is drawn at random from a well-shuffled deck of 52 cards. In which of the following cases are the events E and F independent?
(i) E: ‘the card drawn is a spade’
F: ‘the card drawn is an ace ’
(ii) E: ‘the card drawn is black’
F: ‘the card drawn is a king’
(iii) E: ‘the card drawn is a king or queen ’
F: ‘the card drawn is a queen or jack ’.
Solution:

Question 16
In a hostel, 60% of the students read Hindi newspaper, 40% read English newspaper and 20% read both Hindi and English newspapers. A student is selected at random
(a) Find the probability that she reads neither Hindi nor English newspapers.
(b) If she reads Hindi newspaper, find the probability that she reads english newspapers.
(c) If she reads English newspaper, find the probability that she reads Hindi newspaper.
Solution:

Choose the correct answer in the following Question 17 and 18:

Question 17
The probability of obtaining an even prime number on each die when a pair of dice is rolled is
(a) 0
(b) $\frac { 1 }{ 3 }$
(c) $\frac { 1 }{ 12 }$
(d) $\frac { 1 }{ 36 }$
Solution:

Question 18
Two events A and B are said to be independent, if
(a) A and B are mutually exclusive
(b) P(A’B’) = [1 – P(A)] [1 – P(B)]
(c) P(A) = P(B)
(d) P (A) + P (B) = 1
Solution:

NCERT Solutions for Class 12 Maths Chapter 13 Probability 13.3 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Probability Ex 13.3

Chapter 13 Probability Exercise 13.3

Question 1.
An urn contain 5 red and 5 black balls. A ball is drawn at random, its colour is noted and is returned to the urn. Moreover, 2 additional balls of the colour drawn are put in the urn and then a ball is drawn at random.What is the probability that the second ball is red?
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.3 Q 1

Question 2.
A bag contains 4 red and 4 black balls, another bag contains 2 red and 6 black balls. One of the two bags is selected at random and a ball is drawn from the bag which is found to be red. Find the probability that the ball is drawn from the first bag.
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.3 Q 2

Question 3.
Of the students In a college, it is known that 60% reside In hostel and 40% are day scholars (not residing In hostel). Previous year results report that 30% of all students who reside in hostel attain A grade and 20% of day scholars attain A grade in their annual examination. At the end of the year, one student Is chosen at random from the college and he has an A- grade what Is the probability that the student is a hostlier?
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.3 Q 3

Question 4.
In answering a question on a multiple choice test, a student either knows the answer or 3 guesses. Let $\frac { 3 }{ 4 }$ be the probability that he knows the answer and $\frac { 1 }{ 4 }$ be the probability that he guesses. Assuming that a student who guesses at the answer will be correct with probability $\frac { 1 }{ 4 }$ . What is the probability that the student knows the answer given that he answered it correctly?
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.3 Q 4

Question 5.
A laboratory blood test is 99% effective in detecting a certain disease when it is, in fact, present. However, the test also yields a false positive result for 0.5% of the healthy person tested (i.e. if a healthy person is tested, then, with probability 0.005, the test will imply he has the disease). If 0.1 percent of the population actually has the disease, what is the probability that a person has the disease given that his test result is positive?
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.3 Q 5

Question 6.
There are three coins. One is a two headed coin, another is a biased coin that conies up heads 75% of the time and third is an unbiased coin. One of the three coins is choosen at random and tossed, it shows head, what is the probability that it was the two headed coin?
Solution:
Probability Class 12 Maths NCERT Solutions Ex 13.3 Q 6

Question 7.
An insurance company insured 2000 scooter drivers, 4000 car drivers and 6000 truck drivers. The probability of mi accident are 0.01, 0.03, 0.15 respectively. One of the insured persons meets with an accident. What is the probability that he is a scooter driver?
Solution:
Probability Class 12 Maths NCERT Solutions Ex 13.3 Q 7

Question 8
A factory has two machines A and B. Past record shows that machine A produced 60% of the items of output and machine B produced 40% of the items. Further, 2% of the items produced by machine A and 1% produced by machine B were defective.All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B.?
Solution:
Probability Class 12 Maths NCERT Solutions Ex 13.3 Q 8

Question 9.
Two groups are competing for the position on the Board of directors of a corporation. The probabilities that the first and the second groups will win are 0.6 and 0.4 respectively. Further, if the first group wins, the probability of introducing a new product is 0.7 and the corresponding probability is 0.3 if the second group wins. Find the probability that the new product introduced was by the second group.
Solution:
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.3 Q 9

Question 10.
Suppose a girl throws a die. If she gets a 5 or 6, she tosses a coin three times and notes the number of heads.If she gets 1,2,3 or 4, she tosses a coin once and notes whether a head or tail is obtained. If she obtained exactly one head, what is the probability that she threw 1,2,3, or 4 with the die?
Solution:
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.3 Q 10

Question 11.
A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?
Solution:
Probability Class 12 Maths NCERT Solutions Ex 13.3 Q 11

Question 12.
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 both diamonds. Find the probability of the lost card being a diamond?
Solution:
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.3 Q 12

Question 13.
Probability that A speaks truth is 4/5. A coin is tossed. A reports that a head appears. The probability that actually there was head is
A. 4/5
B. 1/2
C.1/5
D.2/5
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.3 Q 13

Question 14.
If A and B are two events such that A⊂B and P (B) ≠ 0, then which of the following is correct:
(a) P(A | B) = $\frac { P(B) }{ P(A) }$
(b) P (A | B) < P (A)
(c) P(A | B) ≥ P(A)
(d) None of these
Probability Maths Class 12 NCERT Solutions Ex 13.3 Q 14

NCERT Solutions for Class 12 Maths Chapter 13 Probability 13.4 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Probability Ex 13.4

Question 1.
State which of the following are not the probability distributions of a random variable. Give reasons for your answer.
NCERT Solutions for Class 12 Maths Chapter 13 Probability 1
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.4 Q 1

Question 2.
An urn contains 5 red and 2 black balls. Two balls are randomly drawn. Let X represent the number of black balls. What are the possible values of X ?. Is X a random variable?
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.4 Q 2

Question 3.
Let X represent the difference between the number of heads and the number of tails obtained when a coin is tossed 6 times. What are possible values of X?
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.4 Q 3

Question 4.
Find the probability distribution of
(a) number of heads in two tosses of a coin.
(b) number of tails in the simultaneous tosses of three coins.
(c) number of heads in four tosses of a coin.
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.4 Q 4

Question 5.
Find the probability distribution of the number of successes in two tosses of a die where a success Is defined as
(i) number greater than 4
(ii) six appears on at least one die
Solution:
Maths NCERT Solutions Chapter 13 Probability Ex 13.4 Q 5

Question 6.
From a lot of 30 bulbs which include 6 defectives, a sample of 4 bulbs is drawn at random with replacement Find the probability distribution of the number of defective bulbs.
Solution:
Maths NCERT Solutions Chapter 13 Probability Ex 13.4 Q 6

Question 7.
A coin is biased so that the head is 3 times as likely to occur as tail. If the coin is tossed twice, find the probability distribution of number of tads.
Solution:
Maths NCERT Solutions Chapter 13 Probability Ex 13.4 Q 7

Question 8
A random variable X has the following probability distribution:
NCERT Solutions for Class 12 Maths Chapter 13 Probability 8
Determine
(i) k
(ii) P(X<3) (iii) P(X>6)
(iv) P(0<X<3)
Solution:
Maths NCERT Solutions Chapter 13 Probability Ex 13.4 Q 8

Question 9.
The random variable X has a probability distribution P (X) of the following form, where k is some number
NCERT Solutions for Class 12 Maths Chapter 13 Probability 9.1
(a) Determine the value of k
(b) FindP(X<2),P (X≤2), P(X≥2)
Solution:
Maths NCERT Solutions Chapter 13 Probability Ex 13.4 Q 9

Question 10.
Find the mean number of heads in three tosses of a fair coin.
Solution:
Maths NCERT Solutions Chapter 13 Probability Ex 13.4 Q 10

Question 11.
Two dice are thrown simultaneously. If X denotes the number of sixes, find the expectation of X.
Solution:
NCERT Solutions for Class 12 Maths Chapter 13 Ex 13.4 Q 11

Question 12.
Two numbers are selected at random (without replacement) from the first six positive integers. Let X denote the larger of the two numbers obtained. Find E (X)
Solution:
NCERT Solutions for Class 12 Maths Chapter 13 Ex 13.4 Q 12

Question 13.
Let X denote the sum of the numbers obtained when two fair dice are rolled. Find the variance and standard deviation of X.
Solution:
NCERT Solutions for Class 12 Maths Chapter 13 Ex 13.4 Q 13

Question 14.
A class has 15 students whose ages are 14,17, 15,14,21,17,19,20,16,18,20,17,16,19 and 20 years. One student is selected in such a manner that each has the same chance of being chosen and the age X of the selected student is recorded.What is the probability distribution of the random variable X ? Find mean, variance and standard deviation of X?
Solution:

NCERT Solutions Class 12 Maths Chapter 13 Ex 13.4 Q 14 - i

Question 15.
In a meeting, 70% of the members favour and 30% oppose a certain proposal. A member is selected at random and we take X = 0, if he opposed, and X = 1, if he is in favour Find E (X) and Var (X).
Solution:
NCERT Solutions Class 12 Maths Chapter 13 Ex 13.4 Q 15

Choose the correct answer in each of the following:

Question 16.
The mean of the number obtained on throwing a die having written 1 on three faces, 2 on two faces and 5 on one face is
(a) 1
(b) 2
(c) 5
(d) $\frac { 8 }{ 3 }$
Solution:
Probability Class 12 Maths NCERT Solutions Chapter 13 Ex 13.4 Q 16

Question 17.
Suppose that two cards are drawn at random from a deck of cards. Let X be the number of aces obtained. What is the value of E (X)?
(a) $\frac { 37 }{ 221 }$
(b) $\frac { 5 }{ 13 }$
(c) $\frac { 1 }{ 13 }$
(d) $\frac { 2 }{ 13 }$
Solution:
Probability Class 12 Maths NCERT Solutions Chapter 13 Ex 13.4 Q 17

NCERT Solutions for Class 12 Maths Chapter 13 Probability 13.5 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Probability Ex 13.5

Question 1.
A die is thrown 6 times. If ‘getting an odd number’ is a success, what is the probability of
(i) 5 successes?
(ii) at least 5 successes?
(iii) at most 5 successes?
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.5 Q 1

Question 2.
There are 5% defective items in a large bulk of items. What is the probability that a sample of 10 items will include not more than one defective item ?
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.5 Q 2

Question 3.
A pair of dice is thrown 4 times. If getting a doublet is considered a success, find the probability, of two successes.
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.5 Q 3

Question 4.
Five cards are drawn successively with replacement from a well- shuffled deck of 52 cards. What is the probability that
(i) all the five cards are spades?
(ii) only 3 cards are spades?
(iii) none is spade?
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.5 Q 4

Question 5.
The probability that a bulb produced by a factory will fuse after 150 days of use is 0.05. Find the probability that out of 5 such bulbs.
(i) none
(ii) not more than one
(iii) more than one
(iv) at least one will fuse after 150 days of use
Solution:
Class 12 Maths NCERT Solutions Chapter 13 Probability Ex 13.5 Q 5

Question 6.
A bag consists of 10 balls each marked with one of the digits 0 to 9. If four bails are drawn successively with replacement from the bag, what is the probability that none is marked with the digit 0?
Solution:
Probability Class 12 Maths NCERT Solutions Chapter 13 Ex 13.5 Q 6

Question 7.
In an examination, 20 questions of true – false type are asked. Suppose a student tosses fair coin to determine his answer to each question. If the coin falls heads, he answers ‘true,’ if it falls tails, he answers “ false’. Find the probability that he answers at least 12 questions correctly.
Solution:
Probability Class 12 Maths NCERT Solutions Chapter 13 Ex 13.5 Q 7

Question 8
Suppose X has a binomial distribution $B\left( 6,\frac { 1 }{ 2 } \right)$ . Show that X = 3 is the most likely outcome.
(Hint: P (X = 3) is the maximum among all P (Xi), xi. = 0,1,2,3,4,5,6)
Solution:
Probability Class 12 Maths NCERT Solutions Chapter 13 Ex 13.5 Q 8

Question 9.
On a multiple choice examination with three possible answers for each of the five questions, what is the probability that a candidate would get four or more correct answers just by guessing?
Solution:
Probability Class 12 Maths NCERT Solutions Chapter 13 Ex 13.5 Q 9

Question 10.
A person buys a lottery ticket in 50 lotteries, in each of which his chance of winning a prize is $\frac { 1 }{ 100 }$ . What is the probability that he will win a prize?
(a) at least once,
(b) exactly once,
(c) at least twice?
Solution:
Probability Class 12 Maths NCERT Solutions Chapter 13 Ex 13.5 Q 10

Question 11.
Find the probability of getting 5 exactly twice in 7 throws of a die.
Solution:
NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.5 Q 11

Question 12.
Find the probability of throwing at most 2 sixes in 6 throws of a single die.
Solution:
Probability Class 12 Maths NCERT Solutions Chapter 13 Ex 13.5 Q 12

Question 13.
It is known that 10% of certain articles manufactured are defective. What is the probability that in a random sample of 12 such articles 9 are defective?
Solution:
Probability Class 12 Maths NCERT Solutions Chapter 13 Ex 13.5 Q 13

Question 14.
In a box containing 100 bulbs, 10 are defective. The probability that out of a sample of 5 bulbs, none is defective is
(a) ${ 10 }^{ -1 }$
(b) ${ \left( \frac { 1 }{ 2 } \right) }^{ 5 }$
(c) ${ \left( \frac { 9 }{ 10 } \right) }^{ 5 }$
(d) $\frac { 9 }{ 10 }$
Solution:
Probability Class 12 Maths NCERT Solutions Chapter 13 Ex 13.5 Q 14

Question 15.
The probability that a student is not a swimmer is $\frac { 1 }{ 5 }$ . Then the probability that out of five students, four are swimmers is:
NCERT Solutions for Class 12 Maths Chapter 13 Probability 15
Solution:
NCERT Solutions Class 12 Maths Chapter 13 Probability Ex 13.5 Q 15

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NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.1

NCERT Solutions for Class 12 Maths Chapter 13 Probability Ex 13.2

Chapter 13 Probability Exercise 13.3

NCERT Solutions for Class 12 Maths Chapter 13 Probability 13.4 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Probability Ex 13.4

NCERT Solutions for Class 12 Maths Chapter 13 Probability 13.5 are part of NCERT Solutions for Class 12 Maths. Here we have given Class 12 Maths NCERT Solutions Probability Ex 13.5

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