question_answer

Let A, B and C be three events, which are pair-wise independence and\[\bar{E}\]denotes the complement of an event E.\[P(A\cap B\cap C)=0\]and \[P(C)>0,\]then \[P[(\bar{A}\cap \bar{B}|C)]\]is equal to [JEE Main 16-4-2018]

A) \[P(A+P(\bar{B})\]                

B) \[P(\bar{A})-P(\bar{B})\]

C) \[P(\bar{A})-P(B)\]                 

D) \[P(\bar{A})+P(\bar{B})\]    

Correct Answer: C

Solution :

 We need find \[P(\bar{A}\cap \bar{B}\cap |C)=\]shaded portions in Venn Diagram \[=P(\bar{A}\cap \bar{B}\cap |C)=\frac{P(\bar{A}\cap \bar{B}\cap C)}{P(C)}\] \[=\frac{P(C)-P(A\cap C-P(B\cap C)}{P(C)}\] \[=-\frac{P(A).P(C)-P(B).P(C)}{P(C)}\] \[=1-P(A)-P(B)\] \[=P(\bar{A})-P(B)\]