Question:
If $P(A \mid B)>P(A)$, then which of the following is correct:
(A) $P(B \mid A) (B) $P(A \cap B) (C) $P(B \mid A)>P(B)$ (D) $P(B \mid A)=P(B)$
Solution:
$P(A \mid B)>P(A)$
$\Rightarrow \frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})}>\mathrm{P}(\mathrm{A})$
$\Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})>\mathrm{P}(\mathrm{A}) \cdot \mathrm{P}(\mathrm{B})$
$\Rightarrow \frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{A})}>\mathrm{P}(\mathrm{B})$
$\Rightarrow \mathrm{P}(\mathrm{B} \mid \mathrm{A})>\mathrm{P}(\mathrm{B})$c
Thus, the correct answer is C.
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