## Book: Mathematics Part-II

### Chapter: 13. Probability

#### Subject: Maths - Class 12th

##### Q. No. 18 of Exercise 13.2

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##### Two events A and B will be independent, if

Given: Two events A and B will be independent.

As A and B are independent events.

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

We solve it using options.

Let P(A) = a, P(B) = b

As, A and B are mutually exclusive

P (A B) = ϕ

Now, P(A).P(B) = a.b ≠ P(A B)

P (A B) P(A) . P(B)

Hence, it shows A and B are not Independent events.

(B) P(A′B′) = [1 – P(A)] [1 – P(B)]

P(A′ B) = 1 P(A) P(B) + P(A)P(B)

1 - P (A B) =1 – P(A) – P(B) + P(A)P(B)

= - [P(A) + P(B) - P (A B)] = P(A) P(B) + P(A)P(B)

= - P(A) - P(B) + P (A B) = P(A) P(B) + P(A)P(B)

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

Hence, it shows A and B are Independent events.

(C) P(A) = P(B)

As, P(A) = P(B)

Let we take the example of a coin

P(A) = probability of getting head = 1/2

P(B) = probability of getting tail = 1/2

A B = ϕ

P(A B) = probability of getting head and tail both = 0

Now, P(A).P(B) = 1/2 . 1/2 = 1/4 ≠ P(A B)

P (A B) P(A) . P(B)

Hence, it shows A and B are not Independent events.

(D) P(A) + P(B) = 1

Let we take the example of a coin

P(A) = probability of getting head = 1/2

P(B) = probability of getting tail = 1/2

Now, P(A) + P(B) = 1/2 + 1/2 = 1

But it doesnot inferred that A and B are independent.

Hence the correct option is B.

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