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If $\mathrm{A}$ and $\mathrm{B}$ are two events such that $2 \mathrm{P}(\mathrm{A})=3 \mathrm{P}(\mathrm{B})$, where 0 $ < \mathrm{P}(\mathrm{A}) < \mathrm{P}(\mathrm{B}) < 1$, then which one of the following is correct?
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The correct answer is:
$\mathrm{P}(\mathrm{A} \cap \mathrm{B}) < \mathrm{P}(\mathrm{B} \mid \mathrm{A}) < \mathrm{P}(\mathrm{A} \mid \mathrm{B})$
Given, $2 . \mathrm{P}(\mathrm{A})=3 . \mathrm{P}(\mathrm{B})$
$\Rightarrow \frac{2 \mathrm{P}(\mathrm{A})}{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}=\frac{3 \mathrm{P}(\mathrm{B})}{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}$
$\Rightarrow \frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{2 \cdot \mathrm{P}(\mathrm{A})}=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{3 \mathrm{P}(\mathrm{B})}$
$\Rightarrow \frac{1}{2} \cdot \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)=\frac{1}{3} \cdot \mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)$
$\Rightarrow \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right) < \mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right) .$
$\Rightarrow \frac{2 \mathrm{P}(\mathrm{A})}{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}=\frac{3 \mathrm{P}(\mathrm{B})}{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}$
$\Rightarrow \frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{2 \cdot \mathrm{P}(\mathrm{A})}=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{3 \mathrm{P}(\mathrm{B})}$
$\Rightarrow \frac{1}{2} \cdot \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)=\frac{1}{3} \cdot \mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right)$
$\Rightarrow \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right) < \mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right) .$
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