Question

Assertion :Let $A,B$ and $C$ be three events such that $A$ is independent of both $B$ and $C$.

$A$ is independent of $B\cup C$.

Reason: $A$ is independent of $B\cap C$.

• A. Both Assertion and Reason are correct and Reason is the correct explanation for Assertion
• B. Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion
• C. Assertion is correct but Reason is incorrect
• D. Assertion is incorrect but Reason is correct

Answer 1

The correct option is B Both Assertion and Reason are correct but Reason is not the correct explanation for Assertion

$P[A\cap (B\cup C\left)\right]=P\left[\right(A\cap B)\cup (A\cap C\left)\right]$
$=P(A\cap B)+P(A\cap C)-P(A\cap B\cap C)$
$=P\left(A\right)P\left(B\right)+P\left(A\right)P\left(C\right)-P(A\cap B\cap C)$
$A$ and $B\cup C$ are independent.
$\Rightarrow P[A\cap (B\cup C\left)\right]=P\left(A\right)P(B\cup C)$
$\Rightarrow P\left(A\right)P\left(B\right)+P\left(A\right)P\left(C\right)-P(A\cap B\cap C)$
$=P\left(A\right)\left[P\right(B)+P(C)-P(B\cap C\left)\right]$
$\Rightarrow P(A\cap B\cap C)=P\left(A\right)P(B\cap C)$
$\Rightarrow$ $A$ and $B\cap C$ are independent.