Probability 3 Question 10
10. Let $E^{c}$ denotes the complement of an event $E$. If $E, F, G$ are pairwise independent events with $P(G)>0$ and $P(E \cap F \cap G)=0$. Then, $P\left(E^{c} \cap F^{c} \mid G\right)$ equals(2007, 3M)
(a) $P\left(E^{c}\right)+P\left(F^{c}\right)$
(b) $P\left(E^{c}\right)-P\left(F^{c}\right)$
(c) $P\left(E^{c}\right)-P(F)$
(d) $P(E)-P\left(F^{c}\right)$
Show Answer
Solution:
- $P \frac{E^{c} \cap F^{c}}{G}=\frac{P\left(E^{c} \cap F^{c} \cap G\right)}{P(G)}$
$$ \begin{aligned} & =\frac{P(G)-P(E \cap G)-P(G \cap F)}{P(G)} \\ & =\frac{P(G)[1-P(E)-P(F)]}{P(G)} \quad[\because P(G) \neq 0] \\ & =1-P(E)-P(F)=P\left(E^{c}\right)-P(F) \end{aligned} $$
