NCERT Solutions class 12 Maths Exercise 13.1

NCERT Solutions class 12 Maths Exercise 13.1 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.

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NCERT Solutions class 12 Maths Probability

1. Given that E and F are events such that P (E) = 0.6, P (F) = 0.3 and P (EF) = 0.02, find P (E) and P.

Ans. Given:  P (E) = 0.6, P (F) = 0.3 and P (E  F) = 0.2

  P =  =

And P =  =

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2. Compute P if P (B) = 0.5 and P (AB) = 0.32

Ans. Given: P (B) = 0.5, P (A  B) = 0.32

 P =  =  = 0.64

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3. If P (A) = 0.8, P (B) = 0.5 and P = 0.4, find:

(i) P (AB) 

(ii) P 

(iii) P (AB)

Ans. (i) P (A  B) = P . P (A) = 0.4 x 0.8 = 0.32

(ii) P =  =  = 0.64

(iii) P (A  B) = P(A) + P (B) – P (A  B) = 0.8 + 0.5 – 0.32 = 0.98

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NCERT Solutions class 12 Maths Exercise 13.1

4. Evaluate P (AB) if 2P (A) = P (B) =  and P =  

Ans. Given: 2 P (A) = P (B) = , P

 P (A) =

Now P (A  B) = P . P (B) =

And P (A  B) = P(A) + P (B) – P (A  B) =  =

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NCERT Solutions class 12 Maths Exercise 13.1

5. If P (A) =  P (B) =  and P (AB) =

Ans. (i) P (A  B) = P(A) + P (B) – P (A  B)

   P (A  B)

 P (A  B) =

(ii) P =  =

(iii) P =  =

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NCERT Solutions class 12 Maths Exercise 13.1

6. Determine P : A coin is tossed three times.

(i) E : heads 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.

Ans. A coin tossed three times, i.e.,

S = (TTT, HTT, THT, TTH, HHT, HTH, THH, HHH)

   = 8

(i) E : heads on third toss

E = (TTH, HTH, THH, HHH)

 

P (E) =

F : heads on first two tosses

F = (HHT, HHH)

  =2

P (F) =

  E  F = (HHH)

  (E  F) = 1

 P (E  F) =

And P =

(ii) E : at least two heads

E = (HHT, HTH, THH, HHH)

 

P (E) =

F : at most two heads

F = (TTT, HTT, THT, TTH, HHT, HTH, THH)

  = 7

P (F) =

  E  F = (HHT, HTH, THH)

  (E  F) = 3

 P (E  F) =

And P =

(iii) E : at most two tails

E = (HTT, THT, TTH, HHT, HTH, THH, HHH)

 

P (E) =

F : at least one tail

F = (TTT, HTT, THT, TTH, HHT, HTH, THH)

  = 7

P (F) =

  E  F = (HTT, THT, TTH, HHT, HTH, THH)

  (E  F) = 6

 P (E  F) =

And P =

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NCERT Solutions class 12 Maths Exercise 13.1

7. Determine P : 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.

Ans. S = (HH, TH, HT, TT)      = 4

(i) E : tail appears on one coin

E = (TH, HT)    

P (E) =

F : one coin shows head

F = (TH, HT)     = 2

P (F) =

  E  F = (TH, HT)   (E  F) = 2

 P (E  F) =

And P =

(ii) E : no tail appears

E = (HH)     

P (E) =

F : no head appears

F = (TT)      = 1

P (F) =

  E  F =      (E  F) = 0

 P (E  F) =

And P =

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NCERT Solutions class 12 Maths Exercise 13.1

8. Determine P : A dice is thrown three times.

E : 4 appears on the third toss, F : 6 and 5 appears respectively on first two tosses.

Ans. Since a dice has six faces. Therefore  = 6 x 6 x 6 = 216

E = (1, 2, 3, 4, 5, 6) x (1, 2, 3, 4, 5, 6) x (4)

F = (6) x (5) x (1, 2, 3, 4, 5, 6)

  = 1 x 1 x 6 = 6

P (F) =

  E  F = (6, 5, 4)

  (E  F) = 1

 P (E  F) =

And P =

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NCERT Solutions class 12 Maths Exercise 13.1

9. Determine : Mother, father and son line up at random for a family picture.

E : Son on one end, F : Father in middle.

Ans. S = (MFS, MSF, SFM, SMF, FMS, FSM)    

E : Son on one end

E = (MFS, SFM, SMF, FMS)    

F : Father in middle

F = (MFS, SFM)       = 2

P (F) =

  E  F = (MFS, SFM)     (E  F) = 2

 P (E  F) =

And P =

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NCERT Solutions class 12 Maths Exercise 13.1

10. A black 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.

Ans. (a)  = 6 x 6 = 36

Let A represents obtaining a sum greater than 9 and B represents black die resulted in a 5.

A = (46, 64, 55, 36, 63, 45, 54, 65, 56, 66)    = 10

P (A) =

B = (51, 52, 53, 54, 55, 56)      = 6

P (B) =

A  B = (55, 56)       = 2

P (A  B) =

P =

(b) Let A denoted the sum is 8

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

B = Red die results in a number less than 4, either first or second die is red

 B = {(1, 1), (1, 2), (1, 3), (1, 4), (1, 5), (1, 6), (2, 1), (2, 2), (2, 3), (2, 4), (2, 5), (2, 6),

(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)}

P (B) =

A  B = {(2, 6), (3, 5)      = 2

P (A  B) =  =

P =

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NCERT Solutions class 12 Maths Exercise 13.1

11. A fair die is rolled. Consider events E = {1, 3, 5}, F = {2, 3} and G = {2, 3, 4, 5}. Find:

(i) P  and P

(ii) P  and P

(iii) P  and P

Ans. S = (1, 2, 3, 4, 5, 6)       = 6

E = (1, 3, 5)  F = (2, 3)  G = (2, 3, 4, 5)

  = 3   = 2   = 4

(i) P (E) =     P (F) =

E  F = (3)      = 1

P (E  F) =

P =  and P =

(ii) P (E) =     P (G) =

E  G = (3, 5)      = 2

P (E  G) =

P =  and P =

(iii) P (G) =

E  F = (1, 2, 3, 5) and G (2, 3, 4, 5)

(E  F)  G = (2, 3, 5)    = 3

P =

Again E  F = (3)

(E  F)  G = (3)    = 1

P =

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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 a girl?

Ans. Let first and second girl are denoted by G₁ and G₂ and boys B₁ and B₂.

  S = {(G₁G₂), (G₁B₂), (G₂B₁), (B₁B₂)}

Let A = Both the children are girls = (G₁G₂)

B = Youngest child is girl = {(G₁G₂), (B₁G₂)}

C = at least one is a girl = {(G₁B₂), (G₁G₂), (B₁G₂)}

A  B = (G₁G₂)       P (A  B) =

A  C = (G₁G₂)       P (A  C) =

P (B) =  and P (C) =

(i) P =

(ii) P =

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13. An instructor has a test bank consisting of 300 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 test bank, what is the probability that it will be an easy question given that it is a multiple choice question?

Ans. Number of easy True/False questions = 300

Number of difficult True/False questions = 200

Number of easy multiple choice questions = 500

Number of difficult multiple choice questions = 400

Total number of all such questions =  = 1400

Let E represents an easy question and F represents a multiple choice question.

   = 300 + 500 = 800 and  = 500 + 400 = 900

P (F) =

 = 500   P (E  F) =

P =

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NCERT Solutions class 12 Maths Exercise 13.1

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’.

Ans. S =  (1, 1), (2, 1), (3, 1), (4, 1), (5, 1), (6, 1)

(1, 2), (2, 2), (3, 2), (4, 2), (5, 2), (6, 2)

(1, 3), (2, 3), (3, 3), (4, 3), (5, 3), (6, 3)

(1, 4), (2, 4), (3, 4), (4, 4), (5, 4), (6, 4)

(1, 5), (2, 5), (3, 5), (4, 5), (5, 5), (6, 5)

(1, 6), (2, 6), (3, 6), (4, 6), (5, 6), (6, 6)

   = 36

Let A represents the event “the sum of numbers on the dice is 4” and B represents the event “the two numbers appearing on throwing two dice are different”.

Therefore, A = {(1, 3), (2, 2), (3, 1)}   = 3

P (A) =

Also B = {(2, 1), (3, 1), (4, 1), (5, 1), (6, 1), (1, 2), (3, 2), (4, 2), (5, 2), (6, 2),(1, 3), (2, 3), (4, 3), (5, 3), (6, 3), (1, 4), (2, 4), (3, 4), (5, 4), (6, 4),(1, 5), (2, 5), (3, 5), (4, 5), (6, 5), (1, 6), (2, 6), (3, 6), (4, 6), (5, 6)}

 = 30

P (B) =

Now A  B = {(1, 3), (3, 1)}

  = 2

P (A  B) =

Hence, P =

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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”.

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

(1, H), (2, H), (3, H), (4, H), (5, H), (1, T), (2, T), (3, T), (4, T), (5, T)}

   = 20

P (first die shows a multiple of 3) =

P (first die shows a number which is not a multiple of 3) =

Let A = the coin shows a tail = {(1, T), (2, T), (4, T), (5, T)}

B = at least one die shows a 3 = {(3, 1), (3, 2), (3, 3), (3, 4), (3, 5), (3, 6)}

A  B =

 = 4,  = 6,  = 0

P (B) =   and P (A  B) = 0

P =  =

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NCERT Solutions class 12 Maths Exercise 13.1

In each of the following choose the correct answer:

16. If P (A) =  P (B) = 0, then P is:

(A) 0  

(B)     

(C) not defined

(D) 1

Ans. P (A) =  P(B) = 0

   = 0

 P =  =  = not defined

Therefore, option (C) is correct.

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NCERT Solutions class 12 Maths Exercise 13.1

17. If A and B are events such that P = P, then:

(A) AB 

(B) A = B 

(C) AB =    

(D) P (A) = P (B)

Ans. P = P

  

 P (A) = P (B)

Therefore, option (D) is correct.

NCERT Solutions class 12 Maths Exercise 13.1

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