Maths · Probability: Probability of an event, addition and multiplication theorems of probability
If =\boldsymbol{P}(\boldsymbol{B} / \boldsymbol{A}) . \boldsymbol{A} \) and are
If \( \boldsymbol{P}(\boldsymbol{A} / \boldsymbol{B})=\boldsymbol{P}(\boldsymbol{B} / \boldsymbol{A}) . \boldsymbol{A} \) and \( \boldsymbol{B} \) are two non-mutually exclusive events then
- A. \( A \) and \( B \) are necessarily same events
- B. \( P(A)=P(B) \)
- C. \( P(A \cap B)=P(A) P(B) \)
- D. all the above
Step-by-step solution
Given P(A|B) = P(B|A). By definition, P(A|B) = P(A∩B)/P(B) and P(B|A) = P(A∩B)/P(A). Equating gives P(A∩B)/P(B) = P(A∩B)/P(A). Since A and B are non-mutually exclusive, P(A∩B) ≠ 0, so cancel to get 1/P(B) = 1/P(A), thus P(A) = P(B). Option A is false because events need not be identical; Option C is false as independence does not follow; Option D is false. Hence only B is correct.
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