Question

If $A,B,C$ are mutually independent events, such that $0<P\left(A\right),P\left(B\right),P\left(C\right)<1,$ then which of the following statement(s) is/are correct ?
$I.A$ and $B\cup C$ are independent events.
$II.A$ and $B\cap C$ are independent events.

• A. Only $I$ is true
• B. Only $II$ is true
• C. Both are true
• D. Both are false

Answer 1

The correct option is C Both are true

$I.$
$P[A\cap (B\cup C\left)\right]=P\left(A\right)+P(B\cup C)-P(A\cup B\cup C)$
$=P\left(A\right)+P\left(B\right)+P\left(C\right)-P(B\cap C)-\left[P\right(A)+P(B)+P(C)-P(A\cap B)-P(B\cap C)-P(A\cap C)+P(A\cap B\cap C\left)\right]$
$=P(A\cap B)+P(A\cap C)-P(A\cap B\cap C)$
$=P\left(A\right).P\left(B\right)+P\left(A\right).P\left(C\right)-P\left(A\right).P\left(B\right).P\left(C\right)$
$=P\left(A\right)\left[P\right(B)+P(C)-P(B).P(C\left)\right]$
$=P\left(A\right).P(B\cup C)$

$II.$
$P[A\cap (B\cap C\left)\right]=P(A\cap B\cap C)=P\left(A\right).P\left(B\right).P\left(C\right)=P\left(A\right).P(B\cap C)$
So, both the statements are true.