Question

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

• A. both $S_1$ and $S_2$ are true
• B. only $S_1$ is true
• C. only $S_2$ is true
• D. neither $S_1$ nor $S_2$ is true

Answer 1

Answer: (A)

both $S_1$ and $S_2$ are true

We are given that
$P(A\cap B)=P\left(A\right)P\left(B\right)$
$P(B\cap C)=P\left(B\right)P\left(C\right),P(C\cap A)=P\left(C\right)P\left(A\right)$, and $P(A\cap B\cap C)=P\left(A\right)P\left(B\right)P\left(C\right)$
We have
$P(A\cap B\cap C)=P(A\cap B\cap C)$
$=P\left(A\right)P\left(B\right)P\left(C\right)=P\left(A\right)P(B\cap C)$.
$\Rightarrow$ $A$ and $B\cap C$ are independent. Therefore, $S_2$ is true.
Also $P\left[\right(A\cap (B\cup C)]=P[(A\cap B)\cup (A\cap C)]$
$=P(A\cap B)+(A\cap C)-P\left[\right(A\cap B)\cap (A\cap C\left)\right]$
$=P(A\cap B)+P(A\cap C)-P(A\cap B\cap C)$
$=P\left(A\right)P\left(B\right)+P\left(A\right)P\left(C\right)-P\left(A\right)P\left(B\right)P\left(C\right)$
$=P\left(A\right)\left[P\right(B)+P(C)-P(B\left)P\right(C\left)\right]$
$=P\left(A\right)\left[P\right(B)+P(C)=P(B\cap C\left)\right]=P\left(A\right)P(B\cup C)$
$\therefore$ $A$ and $B\cup C$ are independent.