Question

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

• A. $P\left(A\right)+P\left(\overline{B}\right)$
• B. $P\left(\overline{A}\right)-P\left(\overline{B}\right)$
• C. $P\left(\overline{A}\right)-P\left(B\right)$
• D. $P\left(\overline{A}\right)+P\left(\overline{B}\right)$

Answer 1

The correct option is C $P\left(\overline{A}\right)-P\left(B\right)$

we need find $P(\overline{A}\cap \overline{B}|C)$ = shaded portions in Venn Diagram

=$P(\overline{A}\cap \overline{B}|C)=\frac{P(\overline{A}\cap \overline{B}\cap C)}{P\left(C\right)}$

=$\frac{P\left(C\right)-P(A\cap C)-P(B\cap C)}{P\left(C\right)}$

=$1-\frac{P\left(A\right).P\left(C\right)+P\left(B\right).P\left(C\right)}{P\left(C\right)}$

=$1-P\left(A\right)-P\left(B\right)$

=$P\left(\overline{A}\right)-P\left(B\right)$