Let A,BandC be three events, which are pair-wise independent and ¯¯¯¯E denotes the complement of an event E. If P(A∩B∩C)=0, then P[(¯¯¯¯A∩¯¯¯¯B)|C] is equal to:
A
P(¯¯¯¯A)−P(B)
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B
P(¯¯¯¯A)−P(¯¯¯¯B)
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C
P(¯¯¯¯A)+P(¯¯¯¯B)
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D
P(A)+P(¯¯¯¯B)
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Solution
The correct option is AP(¯¯¯¯A)−P(B) P[(¯¯¯¯A∩¯¯¯¯B)|C]=P(¯¯¯¯A∩¯¯¯¯B)∩C)P(C)=P((A∪B)′∩C)P(C)=P(C)−P((A∪B)∩C)P(C)[∵P(A∩B′)=P(A)−P(A∩B)]=P(C)−[P(A∪B)+P(C)−P(A∪B∪C)]P(C)=P(C)−[P(A)+P(B)−P(A∩B)+P(C)−P(A∪B∪C)]P(C)=P(C)−[P(A∩C)+P(B∩C)−P(A∩B∩C)]P(C)=P(C)−P(A)P(C)−P(B)P(C)P(C)=1−P(A)−P(B)=P(¯¯¯¯A)−P(B)orP(¯¯¯¯B)−P(A)