Question

Let A, B and C be three events, which are pair-wise independent and $\bar{E}$ denotes the complement of an event E. If $P(A \cap B \cap C) = 0$ and $P(C) > \,0$, then $P [ ( (\bar{A} \cap \bar{B}) | C]$ is equal to :

• A. $P(\bar{A} ) - P(B)$
• B. $P({A} ) + P(\bar{B})$
• C. $P(\bar{A} ) - P(\bar{B})$
• D. $P(\bar{A} ) + P(\bar{B})$

Answer 1

Answer: (A)

$P\left[(\bar{A}\cap\bar{B})|C\right]=\frac{P\left[\left(\bar{A}\cup\bar{B}\right)\cap C\right]}{P\left(C\right)}$
$=\frac{P\left(C\right)-P\left(A\cap C\right)-P\left(B\cap C\right)+P\left(A\cap B\cap C\right)}{P\left(C\right)}$
$=\frac{P\left(C\right)-P\left(A\right)P\left(C\right)+P\left(B\right)P\left(C\right)}{P\left(C\right)}$
$= 1 -P \left(A\right) - P\left(B\right)$
$=P\left(\bar{A}\right)-P\left(B\right)$ or $P\left(\bar{B}\right)-P\left(A\right)$