Probability Question 303
Question: Let $ E^{c} $ denote the complement of an event E. let E, F, G be pairwise independent events with $ P(G)>0 $ and $ P(E\cap F\cap G)=0 $ . Then $ P(E^{c}\cap F^{c}/G) $ equals
Options:
A) $ P(E^{c})+P(F^{c}) $
B) $ P(E^{c})-P(F^{c}) $
C) $ P(E^{c})-P(F) $
D) $ P(E)-P(F^{c}) $
Show Answer
Answer:
Correct Answer: C
Solution:
We have
$ \therefore E\cap F\cap G=\phi $
$ P(E^{c}\cap F^{c}/G)=\frac{P(E^{c}\cap F^{c}\cap G)}{P(G)} $
$ =\frac{P(G)-P(E\cap G)-P(G\cap F)}{P(G)} $ [From Venn diagram $ E^{c}\cap F^{c}\cap G=G-E\cap G-F\cap G $ ] $ =\frac{P(G)-P(E)P(G)-P(G)p(F)}{P(G)} $
$ =1-P(E)-P(F)=P(E^{c})-P(F) $ [ $ \therefore $ E, F, G are pairwise independent]
BETA
