Question

Let $E^C$ denote the complement of an event E. let E, F, G be pair wise independent events with P(G) > 0 and $P(E\cap F\cap G)=0$. Then $P\left(\frac{E^C\cap F^C)}{G}\right)$ equals

• A.
• B.
• C.
• D.

Answer 1

The correct option is C


We have,
$E\cap F\cap G=\phi P(E^C\cap \frac{F^C}{F})=\frac{P(E^C\cap F^C\cap G)}{P\left(G\right)}=\frac{P\left(G\right)-P(E\cap G)-P(G\cap F)}{P\left(G\right)}$
[From Venn diagram $E^C\cap F^C\cap F=G-E\cap G-F\cap G]$
$=\frac{P\left(G\right)-P\left(E\right)P\left(G\right)-P\left(G\right)\left(F\right)}{P\left(G\right)}$
[$\therefore$ E, F, G are pair wise independent]
$=1-P\left(E\right)-P\left(F\right)=P\left(E^C\right)-P\left(F\right)$