**NCERT Solutions for Maths Class 12 Exercise 13.2**

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Class 12th is a very crucial stage of your student’s life, since you take all important decisions about your career on this stage. Mathematics plays a vital role to take decision for your career because if you are good in mathematics, you can choose engineering and technology field as your career.

**NCERT Solutions for Maths Class 12 Exercise 13.2** helps you to solve each and every problem with step by step explanation which makes you strong in mathematics.

All the schools affiliated with CBSE, follow the NCERT books for all subjects. You can check your syllabus from **NCERT Syllabus for Mathematics Class 12**.

**NCERT Solutions for Maths Class 12 Exercise 13.2** are prepared by the experienced teachers of CBSE board. If you are preparing for JEE Mains and NEET level exams, then it will definitely make your foundation strong.

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**NCERT Solutions for Maths Class 12 Exercise 13.1**

**NCERT Solutions for Maths Class 12 Exercise 13.3**

**NCERT Solutions for Maths Class 12 Exercise 13.4**

**NCERT Solutions for Maths Class 12 Exercise 13.5**

**NCERT Solutions for Maths Class
12 Exercise 13.2**

**Maths Class
12 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:**

We have given
that P(A) = 3/5 and P(B) = 1/5

If A and B are
independent event, then P(A∩B) = P(A).P(B)

=
3/5 × 1/5 = 3/25

Hence,
the value of P(A∩B) is 3/25.

**Maths Class
12 Ex 13.2 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:**

Number of
exhaustive cases = 52

Number of black cards = 26

One black card may be drawn in 26 ways.

∴ Probability of getting a black card
on first draw, P(A) = 26/52 = ½

After drawing one
card, the number of cards left = 51

After drawing a
black card, the number of black cards left = 25

∴ Probability of getting a black card
on second draw, P(B) = 25/51

Thus, the
probability of getting both the cards black = P(A).P(B)

= ½ × 25/51 = 25/102

Hence, the
probability that both the cards drawn are black is 25/102.

**Maths Class
12 Ex 13.2 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:**

S = {12 good
oranges, 3 bad oranges),

n(S) = 15

Probability that
the first orange drawn is good, P(A) = 12/15

Probability that
the second orange drawn is good, P(B) = 11/14

Probability that
the third orange drawn is good, P(C) = 10/13

Thus, the
probability of getting all the oranges good = P(A).P(B).P(C)

= 12/15 × 11/14 × 10/13 = 44/91

Hence, the
probability that the box is approved for sale is 44/91.

**Maths Class
12 Ex 13.2 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:**

When a coin is
thrown, head or tail will occur.

Probability of getting head on the coin, P(A) = ½

When a die is tossed, (1, 2, 3, 4, 5, 6) one of them will appear

∴ Probability of getting 3 on the die,
P(B) = 1/6

P(A).P(B) = ½ ×
1/6 = 1/12

When a die and coin is tossed, total number of cases are

H1, H2, H3, H4, H5, H6

T1, T2, T3, T4, T5, T6

Head and 3 will occur only in 1 way

∴ Probability of getting head on coin
and 3 on die = 1/12, i.e., P(A∩B) = 1/12

Therefore, P(A∩B)
= P(A).P(B)

Hence, A and B
are independent events.

**Maths Class
12 Ex 13.2 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:**

Even numbers on
die are 2, 4, 6.

∴ Probability of getting even number, P(A)
= 3/6 = ½

There are two colours of the die: red and green.

Probability of getting red colour, P(B) = ½

Even number in red colour is 2.

∴ Probability of getting red colour and
even number = P(A∩B) = 1/6

P(A).P(B) = ½ × ½
= ¼

We have, P(A∩B) ≠
P(A).P(B)

Hence, A and B
are not independent.

**Maths Class
12 Ex 13.2 Question 6. **

Let E and F be
the events with P(E) = 3/5, P(F) = 3/10 and P(E∩F) = 1/5.
Are E and F independent?

**Solution:**

We have, P(E)
= 3/5, P(F) = 3/10 and P(E∩F)
= 1/5.

∴ P(E) × P(F) = 3/5 × 3/10 = 9/50

Here, P(E∩F) ≠ P(E) × P(F)

∴ The events A and B are not
independent.

**Maths Class
12 Ex 13.2 Question 7. **

Given that the
events A and B are such that P(A) = ½, P(A∪B) = 3/5 and P(B) = p. Find p if
they are

(i) mutually exclusive

(ii) independent.

**Solution:**

**Maths Class
12 Ex 13.2 Question 8.**

Let A and B be
independent events with 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:**

We have, P(A) =
0.3 and P(B) = 0.4

(i) Since, A and
B are independent events.

Therefore, P(A∩B) = P(A).P(B) = 0.3 × 0.4 =
0.12.

(ii) We know
that, P(A∪B) = P(A) + P(B) – P(A).P(B)

= 0.3 + 0.4 – 0.3 × 0.4 = 0.7 – 0.12 = 0.58.

**Maths Class
12 Ex 13.2 Question 9. **

If A and B are
two events such that P(A) = ¼, P(B) = ½ and P(A∩B) = 1/8, find P(not
A and not B).

**Solution:**

**Maths Class
12 Ex 13.2 Question 10.**

Events A and B
are such that P(A) = ½, P(B) = 7/12 and P(not A or not B)
= ¼. State whether A and B are independent.

**Solution:**

**Maths Class
12 Ex 13.2 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:**

(i) Since, A and B are independent events.

∴ P(A and B) = P(A∩B) = P(A) × P(B)

=
0.3 × 0.6 [ ∵ P(A) = 0.3 and P(B) = 0.6]

∴ P(A and B) = 0.18

**Maths Class
12 Ex 13.2 Question 12. **

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

**Solution:**

We have, S = {1, 2,
3, 4, 5, 6}, n(S) = 6

The probability of getting an odd number in a single throw of die = 3/6 = ½

Similarly, the
probability of getting an even number in a single throw of die = 3/6 = ½

Thus, the
probability of getting an even number three times = ½ × ½ × ½ = 1/8

Therefore, the
probability of getting an odd number at least once

= 1 – Probability
of getting an odd number in none of the throws

= 1 – Probability
of getting an even number thrice

= 1 – 1/8 = 7/8

Hence, the
probability of getting an odd number at least once is 7/8.

**Maths Class
12 Ex 13.2 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) first ball is black and second is red.

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

**Solution:**

S = {10 black
balls, 8 red balls}, n(S) = 18

Let drawing of a red ball be a success.

A = {8 red balls}, n(A) = 8

**Maths Class
12 Ex 13.2 Question 14. **

Probability of
solving specific problem independently by A and B are ½ and 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:**

Probability that
A solves the problem = ½

Probability that A does not solve the problem, P(A) = 1 – ½ = ½

Probability that B solves the problem = 1/3

Probability that B does not solve the problem, P(B) = 1 – 1/3 = 2/3

**Maths Class
12 Ex 13.2 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:**

**Maths Class
12 Ex 13.2 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
newspaper.

(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:**

**Maths Class
12 Ex 13.2 Question 17. **

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

(A) 0

(B) 1/3

(C) 1/12

(D) 1/36

**Solution:**

(D) When a pair
of dice is rolled, then n(S) = 36.

The only even
prime number is 2.

Let E be the event of getting an even prime
number on each die.

Therefore,
E = {2, 2}; n(E) = 1

P(E) = n(E)/n(S)
= 1/36

Hence,
the correct answer is option (D).

**Maths Class
12 Ex 13.2 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:**

(B) Two events A
and B are said to be independent, if P(A∩B) = P(A).P(B)

Thus, P(A’∩B’) =
P(A’).P(B’)

We know that
P(A’) = [1 – P(A)] and P(B’) = [1 – P(B)]

Therefore, P(A’∩B’)
= [1 – P(A)].[1 – P(B)]

Or P(A’B’) = [1 –
P(A)] [1 – P(B)]

Hence, the correct answer is option (B).