Let B be an event such that P(B) > 0. Prove: (a) P(A|B) + P(Ac |B) = 1.

Question

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Answer 1

I assume that by Ac you mean the complement of A, usually written A'.

P(A∩B) + P(A'∩B) = P(B) since (A∩B) ∪ (A'∩B) = B
P(A|B) = P(A∩B)/P(B)
P(A'|B) = P(A'∩B)/P(B)

Therefore, P(A|B) + P(A'|B) = P(B)/P(B) = 1