Probability Question 164
Question: If E and F are independent events such that $ 0<P(E)<1 $ and $ 0<P(F)<1, $ then
[IIT 1989]
Options:
A) E and $ F^{c} $ (the complement of the event F) are independent
B) $ E^{c} $ and $ F^{c} $ are independent
C) $ P( \frac{E}{F} )+P( \frac{E^{c}}{F^{c}} )=1 $
D) All of the above
Show Answer
Answer:
Correct Answer: D
Solution:
$ P(E\cap F)=P(E).P(F) $
Now, $ P(E\cap F^{c})=P(E)-P(E\cap F)=P(E)[1-P(F)]=P(E).P(F^{c}) $
and $ P(E^{c}\cap F^{c})=1-P(E\cup F)=1-[P(E)+P(F)-P(E\cap F) $
$ =[1-P(E)][1-P(F)]=P(E^{c})P(F^{c}) $
Also $ P(E/F)=P(E) $ and $ P(E^{c}/F^{c})=P(E^{c}) $
$ \Rightarrow P(E/F)+P(E^{c}/F^{c})=1. $
BETA
