Question

Let $A,B\text{and}C$ be three events, which are pair-wise independent and $\overline{E}$ denotes the complement of an event $E$. If $P(A\cap B\cap C)=0$, 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. $P\left(\overline{A}\right)+P\left(\overline{B}\right)$
• D. $P\left(\overline{A}\right)-P\left(B\right)$

Answer 1

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

$P\left[\right(\overline{A}\cap \overline{B}\left)|C\right]=\frac{P(\overline{A}\cap \overline{B})\cap C)}{P\left(C\right)}=\frac{P\left(\right(A\cup B{)}^{\prime }\cap C)}{P\left(C\right)}=\frac{P\left(C\right)-P\left(\right(A\cup B)\cap C)}{P\left(C\right)}[\because P(A\cap B^{\prime })=P(A)-P(A\cap B\left)\right]=\frac{P\left(C\right)-\left[P\right(A\cup B)+P(C)-P(A\cup B\cup C\left)\right]}{P\left(C\right)}=\frac{P\left(C\right)-\left[P\right(A)+P(B)-P(A\cap B)+P(C)-P(A\cup B\cup C\left)\right]}{P\left(C\right)}=\frac{P\left(C\right)-\left[P\right(A\cap C)+P(B\cap C)-P(A\cap B\cap C\left)\right]}{P\left(C\right)}=\frac{P\left(C\right)-P\left(A\right)P\left(C\right)-P\left(B\right)P\left(C\right)}{P\left(C\right)}=1-P\left(A\right)-P\left(B\right)=P\left(\overline{A}\right)-P\left(B\right)\text{or}P\left(\overline{B}\right)-P\left(A\right)$