Let `A ,B ,C` be three mutually independent events. Consider the two statements `S_1` and `S_2`.
`S_1`: `A` and `BuuC` are independent
`S_2`: A and `BnnC` are independent. Then,

Options

A. Statement-1 is True, Statement-2 is True, Statement-2 is a correct explanation for Statement-1.
B. Statement-1 is True, Statement-2 is True, Statement-2 is not a correct explanation for Statement-1.
C. Statement-1 is True, Statement-2 is False.
D. Statement-1 is False, Statement-2 is True.

Answer

B

Explanation

IT is given that A, B and C are three mutually, independent events.
` P (A capB) = P (A) P(B),P (B capC), = P(B) P(C)`
` P (C capA) = P(C) P(A) andP (A capB capC) = P(A)P(B)P(C)`
Now,
` P(A cap (B cup C))`
` = P {(A cap B) cup ( A cap C)}`
` P ( cap B)+ P ( A cap C) - P {( A cap B) cap ( A cap C))`
` P ( A cap B)+ P ( A cap C) - P ( A cap B cap C)`
` P (A) P (B) + P(A) P(C) - P (A) P (B) P (C)`
`P (A) { P(B) + P(C) -P ( B cap C) }`
` P (A)P (B cup C)`
A and ` B cup C` are independuct events.
Also, ` (A cap B cap C) - P (A) P (B) P (C) = P(A) P ( cap C)`
` Rightarrow A and B cap C` are independent events.