If A and B are events such that P (A|B) = P(B|A), then (A) A ⊂ B but A ≠ B (B) A = B (C) A ∩ B = Φ (D) P(A) = P(B)

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Solution

It is given that there are two events, A and B such that P( A|B )=P( B|A ).

Conditional probability of event A given that event B has already occurred

P( A|B )= P( A∩B ) P( B ) (1)

Conditional probability of event A given that event B has already occurred is,

P( B|A )= P( B∩A ) P( A ) (2)

It is given that equations (1) and (2) are equal, so it can be written as,

P( A∩B ) P( B ) = P( B∩A ) P( A )

Use the property of probability P( B∩A )=P( A∩B ).

1 P( A ) = 1 P( B ) P( A )=P( B )

Hence, option (D) is correct.

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