Question:
Let A, B and C be three events, which are pair-wise independent and \(\overline{E}\) denotes the completement of an event E. If \(P(A \cap B \cap C) = 0\) and \(P(C) \gt 0\), then \(P\left[(\overline{A} \cap \overline{B}) \mid C\right]\) is equal to :
Let A, B and C be three events, which are pair-wise independent and \(\overline{E}\) denotes the completement of an event E. If \(P(A \cap B \cap C) = 0\) and \(P(C) \gt 0\), then \(P\left[(\overline{A} \cap \overline{B}) \mid C\right]\) is equal to :
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Write \(P(\overline{A}\cap\overline{B}\cap C)\) by inclusion-exclusion over C and use pairwise independence with \(P(A\cap B\cap C)=0\).
Updated On: Nov 11, 2026
- \(P(\overline{A}) - P(B)\)
- \(P(A) + P(\overline{B})\)
- \(P(\overline{A}) - P(\overline{B})\)
- \(P(\overline{A}) + P(\overline{B})\)
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The Correct Option is A
Solution and Explanation
\(P(\overline{A}\cap\overline{B}\cap C) = P(C) - P(A\cap C) - P(B\cap C) + P(A\cap B\cap C)\).
By pairwise independence \(P(A\cap C) = P(A)P(C)\) and \(P(B\cap C) = P(B)P(C)\); also \(P(A\cap B\cap C)=0\).
So \(P(\overline{A}\cap\overline{B}\cap C) = P(C)\,[1 - P(A) - P(B)]\).
Dividing by \(P(C) \gt 0\): \(P(\overline{A}\cap\overline{B}\mid C) = 1 - P(A) - P(B) = P(\overline{A}) - P(B)\).
Answer: (1) \(P(\overline{A}) - P(B)\)
By pairwise independence \(P(A\cap C) = P(A)P(C)\) and \(P(B\cap C) = P(B)P(C)\); also \(P(A\cap B\cap C)=0\).
So \(P(\overline{A}\cap\overline{B}\cap C) = P(C)\,[1 - P(A) - P(B)]\).
Dividing by \(P(C) \gt 0\): \(P(\overline{A}\cap\overline{B}\mid C) = 1 - P(A) - P(B) = P(\overline{A}) - P(B)\).
Answer: (1) \(P(\overline{A}) - P(B)\)
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