NCERT Solutions Class 12 Maths Exercise 13.2

NCERT Solutions Class 12 Maths Exercise 13.2 Class 12 Maths book solutions are available in PDF format for free download. These ncert book chapter wise questions and answers are very helpful for CBSE board exam. CBSE recommends NCERT books and most of the questions in CBSE exam are asked from NCERT text books. Class 12 Maths chapter wise NCERT solution for Maths part 1 and Maths part 2 for all the chapters can be downloaded from our website and myCBSEguide mobile app for free.

Download NCERT solutions for Probability as PDF.

 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)

=

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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) =

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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) =

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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)

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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)

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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.

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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)

   

   

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NCERT Solutions Class 12 Maths Exercise 13.2

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

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NCERT Solutions Class 12 Maths Exercise 13.2

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)

=

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NCERT Solutions Class 12 Maths Exercise 13.2

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)

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NCERT Solutions Class 12 Maths Exercise 13.2

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

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NCERT Solutions Class 12 Maths Exercise 13.2

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)

=

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NCERT Solutions Class 12 Maths Exercise 13.2

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).

=

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NCERT Solutions Class 12 Maths Exercise 13.2

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)

=

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NCERT Solutions Class 12 Maths Exercise 13.2

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.

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NCERT Solutions Class 12 Maths Exercise 13.2

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

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NCERT Solutions Class 12 Maths Exercise 13.2

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.

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NCERT Solutions Class 12 Maths Exercise 13.2

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

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