1. /
2. CBSE
3. /
4. Class 12
5. /
6. Mathematics
7. /
8. NCERT Solutions Class 12...

# NCERT Solutions Class 12 Maths Exercise 13.2

### myCBSEguide App

Download the app to get CBSE Sample Papers 2023-24, NCERT Solutions (Revised), Most Important Questions, Previous Year Question Bank, Mock Tests, and Detailed Notes.

## NCERT Solutions class 12 Maths Probability

1. If P (A) =  and P (B) =  find P (AB) if A and B are independent events.

Ans. As A and B are independent events.

Therefore,  = P (A). P (B)

=

#### 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.

Ans. S = 52 cards   = 52

Two cards are drawn without replacement.

A = {26 black cards}   = 26

P (A) =

P (A and B) = P (A). P (B) =

#### 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.

Ans. S = {12 good oranges, 3 bad oranges}

= 15

P (a box is approved) =

#### 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.

Ans. S = {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6), (T, 1), (T, 2), (T, 3), (T, 4), (T, 5), (T, 6)}

= 12

A = {(H, 1), (H, 2), (H, 3), (H, 4), (H, 5), (H, 6)}   = 6

P (A) =

B = {(H, 3)}  = 1

P (B) =

Now (A  B) = {(H, 3)}  = 1

Again P (A). P (B) =

Therefore,  P (A). P (B)

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

Ans. S =    = 6

A = {2, 4, 6}   = 3

P (A) =

B =    = 3

P (B) =

(A  B) = {2}   = 1

Now P (A). P (B) =

Therefore,  P (A). P (B)

#### 6. Let E and F be events with P (E) =  P (F) =  and P (EF) =  Are E and F independent?

Ans. P (E) =  P (F) =

Now P (E). P (F) =

P (E). P (F)

Therefore, E and F are not independent events.

#### 7. Given that the events A and B are such that P (A) =  P (AB) =  and P (B) =  Find  if they are (i) mutually exclusive, (ii) independent.

Ans. P (A) =   = , P (B) =

(i) A and B are mutually exclusive events.

= P (A) + P (B)

(ii) A and B are independent events.

= P (A) + P (B) – P (A). P (B)

#### 8. Let A and B independent events, P (A) = 0.3 and P (B) = 0.4. Find:

(A) P (AB)

(B) P (AB)

(C) P

(D) P

Ans. P (A) = 0.3, P (B) = 0.4

A and B are independent events.

(i)  P (A). P (B) = 0.3 x 0.4 = 0.12

(ii)  = P (A) + P (B) – P (A). P (B) = 0.03 + 0.4 – 0.3 x 0.4 = 0.7 – 0.12 = 0.58

(iii) P =  =  = P (A) = 0.3

(iv) P =  =  = P (B) = 0.4

#### 9. If A and B are two events such that P (A) =  P (B) =  and P (AB) =  find P (not A and not B).

Ans. P (A) =  P (B) =

P (not A) = 1 – P (A) =

P (not B) = 1 – P (B) =

Now P (A). P (B) =

= P (A). P (B)

Thus, A and B are independent events.

Therefore, ‘not A’ and ‘not B’ are independent events.

Hence, P (not A and not B) = P (not A). P (not B)

=

#### 10. Events A and B are such that P (A) =  P (B) =  and P (not A or not B) =  State whether A and B are independent.

Ans.

P (A) =   and P (B) =

P (A). P (B) =

Therefore,  P (A). P (B)

#### 11. Given two independent events A and B such that P (A) = 0.3, P (B) = 0.6. Find:

(A) P (A and B)

(B) P (A and not B)

(C) P (A or B)

(D) P (neither A nor B)

Ans. P (A) = 0.3, P (B) = 0.6

A and B are independent events.

(i) P (A and B) = P (A). P (B) = 0.3 x 0.6 = 0.18

(ii) P (A and not B) =  = P (A) –  = 0.3 – 0.18 = 0.12

(iii) P (A or B) = P (A) + P (B) – P (A and B) = 0.3 + 0.6 – 0.18 = 0.9 – 0.18 = 0.72

(iv) P (neither A nor B) = P [not (A  B)] = 1 – P (A  B) = 1 – 0.72 = 0.28

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

Ans. S = {1, 2, 3, 4, 5, 6}   = 6

Let A represents an odd number.

A = {1, 3, 5}   = 3

P (A) =

= 1 – P (A) =

= 3

Now P (at least one success) = 1 – P (X = 0)

=

#### 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) first ball is black and second is red.

(iii) one of them is black and other is red.

Ans. S = (10 black balls, 8 red balls)   = = 18

Let drawing of a red ball be a success.

A = {8 red balls}    = 8

P (A) =

And  = 1 – P (A) =

= 2

(i) P (both are red ball) = P (A). P (B) =

(ii) P (first is black ball and second is red) = . P (A) =

(iii) P (one of them is black and other is red) = . P (A) + P (A).

=

#### 14. Probability of solving specific problem independently by A and B are  and  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.

Ans. P (A) =  P (B) = ,  =  and

(i) P (the problem is solved) = 1 – P (the problem is not solved)

=

=

=

(ii) P (exactly one of them solves the problem) = P (A).  +  . P (B)

=

#### 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?

Ans. S = {All the 52 cards}   = 52

(i) E = {13 spades}  = 13

P (E) =

F = {4 aces}   = 4

P (F) =

Now E  F = {An ace of spade}  = 1

=

Also, P(E). P(F) =

Therefore,  = P(E). P(F)

Hence, E and F are independent events.

(ii) E = {26 black cards}   = 26

P (E) =

F = {4 kings}    = 4

P (F) =

Now E  F = {2 black kings}  = 2

=

Also, P(E). P(F) =

Therefore,  = P(E). P(F)

Hence, E and F are independent events.

(iii) E = {4 kings, 4 queens}  = 8

P (E) =

F = {4 queens, 4 jacks}   = 8

P (F) =

Now E  F = {4 queens }  = 4

=

Also, P(E). P(F) =

Therefore,   P(E). P(F)

Hence, E and F are not independent events.

#### 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 newspaper.

(c) If she reads English newspapers, find the probability she reads Hindi newspaper.

Ans. Let A represents the student reading Hindi newspaper and B represents the students reading English newspaper.

P (A) =  P (B) =  and P (A and B) =

(a) P (neither Hindi nor English newspaper is read) = 1 – P (A or B)

= 1 – [P (A) + P (B) – P (A and B)]

=

=

(b)

P

(c) P

#### Choose the correct answer in the following:

17. The probability of obtaining an even prime number on each die when a pair of dice is rolled is:

(A) 0 (B)   (C)  (D)

Ans.  = 36

Let A represents an even prime number on each dice.

A = {22}   = 1

P (A) =

Hence option (D) is correct.

#### 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

Ans. P (A’ and B’)

= [1 – P (A)] . [1 – P (B)]

= P (A’). P (B’)

Hence, option (B) is correct

## NCERT Solutions Class 12 Maths Exercise 13.2

NCERT Solutions Class 12 Maths PDF (Download) Free from myCBSEguide app and myCBSEguide website. Ncert solution class 12 Maths includes text book solutions from both part 1 and part 2. NCERT Solutions for CBSE Class 12 Maths have total 13 chapters. 12 Maths NCERT Solutions in PDF for free Download on our website. Ncert Maths class 12 solutions PDF and Maths ncert class 12 PDF solutions with latest modifications and as per the latest CBSE syllabus are only available in myCBSEguide

## CBSE App for Students

To download NCERT Solutions for class 12 Physics, Chemistry, Biology, History, Political Science, Economics, Geography, Computer Science, Home Science, Accountancy, Business Studies and Home Science; do check myCBSEguide app or website. myCBSEguide provides sample papers with solution, test papers for chapter-wise practice, NCERT solutions, NCERT Exemplar solutions, quick revision notes for ready reference, CBSE guess papers and CBSE important question papers. Sample Paper all are made available through the best app for CBSE students and myCBSEguide website.

### Test Generator

Create question paper PDF and online tests with your own name & logo in minutes.

### myCBSEguide

Question Bank, Mock Tests, Exam Papers, NCERT Solutions, Sample Papers, Notes

### 1 thought on “NCERT Solutions Class 12 Maths Exercise 13.2”

1. Good work for CBSE guide for starting this app student can check his mistakes and knowledge about the question I am very impressed from this app thank you cbse guide. All students who did not interested from any subject can easily understood from this app and you have very impressed