﻿ Let A, B, C, be three mutually independent events. Consider the two statements S1 and S2.                         S1 : A and B ∪ C are independent                         S2 : A and B ∩ C are independent Then : Kaysons Education

# Let A, B, C, Be Three Mutually Independent Events. Consider The Two Statements S1 and S2.                         S1 : A and B ∪ C are Independent                         S2 : A and B ∩ C are Independent Then

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## Question

### Solution

Correct option is

Both S1 and S2 are true

we are given that

P(A ∩ B) = P(AP(B)

P(B ∩ C) = P(BP(C), P(C ∩ A) = P(CP(A)

and          P(A ∩ ∩ C) = P(AP(B), P(C)

we have

P(A ∩ (∩ C)) = P(A ∩ ∩ C) = P(AP(B), P(C) = P(A)P(B∩ C).

⇒         A and B ∩ are independent. Therefore, S2 is true Also

Also        P[(A ∩ (∪ C)] = P[(A ∩ B) ∪ (∩ C)]

= P[(A ∩ BP(∩ C) – P[(A ∩ B) ∩ (∩ C)]

= P(A ∩ B) + P (A ∩ C) – P(A ∩ ∩ C)

= P(AP(B) + P(AP(C) – P(AP(BP(C)

= P(A) [P(B) + P(C) – P(BP(C)]

= P(A) [P(B) + P(C) – P(∩ C)] = P(AP(∪ C)

∴         A and B ∪ C are independent.

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