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Question

Let A,B and C be three events, which are pair-wise independent and ¯¯¯¯E denotes the complement of an event E. If P(ABC)=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 D P(¯¯¯¯A)P(B)
P[(¯¯¯¯A¯¯¯¯B)|C]=P(¯¯¯¯A ¯¯¯¯B) C)P(C)=P((AB)C)P(C)=P(C)P((AB)C)P(C) [P(AB)=P(A)P(AB)]=P(C)[P(AB)+P(C)P(ABC)]P(C)=P(C)[P(A)+P(B)P(AB)+P(C)P(ABC)]P(C)=P(C)[P(AC)+P(BC)P(ABC)]P(C)=P(C)P(A)P(C)P(B)P(C)P(C)=1P(A)P(B)=P(¯¯¯¯A)P(B) or P(¯¯¯¯B)P(A)

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