question_answer

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               

A) \[P({{E}^{c}})+P({{F}^{c}})\]

B) \[P({{E}^{c}})-P({{F}^{c}})\]

C) \[P({{E}^{c}})-P(F)\]

D) \[P(E)-P({{F}^{c}})\]

Correct Answer: C

Solution :

[c] 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]