Question

Let A, B, C be three mutually independent events. Consider the two statements $S_1$ and $S_2$
$S_1$: A and $B\cup C$ are independent
$S_2$ : A and $B\cap C$ are independent Then,

• A. both and are true
• B. only is true
• C. only is true
• D. neither nor is true

Answer 1

The correct option is A both and are true

$P(A\cap (B\cap C)=P[(A\cap B)\cup (A\cap C)]=P(A\cap B)+P(A\cap C)-P(A\cap B\cap C)=P(A\left)P\right(B)+P(A\left)P\right(C)-P(A\left)P\right(B\left)P\right(C)=P(A\left)\right[P\left(B\right)+P\left(C\right)-P(B\cap C)]=P(A\left)P\right(B\cup C)$
Therefore, $S_1$ is true.
$P(A\cap (B\cap C)=P(A\left)P\right(B\left)P\right(C)=P(A\left)P\right(B\cup C)$
Therefore, $S_2$ is also true.