Question

Let ABC, 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

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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