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Question:

Let A and B be two non-null events such that A⊆B. Then, which of the following statements is always correct?

P(A|B)=1

P(A|B)≤P(A)

P(A|B)≥P(A)

P(A|B)=P(B)−P(A)

Solution:

Correct option is C. P(A|B)≥P(A)

P(A|B) = P(A∩B)/P(B) = P(A)/P(B) (as A⊆B implies P(A∩B) = P(A))
Since A⊆B, we have P(B) ≥ P(A). Therefore, 1/P(B) ≤ 1/P(A) if P(A) and P(B) are positive.
Then, P(A|B) = P(A)/P(B) ≥ P(A) * P(A)/P(A) = P(A). Therefore P(A|B) ≥ P(A).