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If $\mathrm{P}(\mathrm{A})=\frac{3}{10}, \mathrm{P}(\mathrm{B})=\frac{3}{5}, \mathrm{P}(\mathrm{A} \cup \mathrm{B})=\frac{3}{5}$, then $\mathrm{P}(\mathrm{A} / \mathrm{B}) \times \mathrm{P}(\mathrm{B} / \mathrm{A})=$
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Verified Answer
The correct answer is:
$\frac{1}{12}$
$$
\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right) \times \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})} \times \frac{\mathrm{P}(\mathrm{B} \cap \mathrm{A})}{\mathrm{P}(\mathrm{A})}
$$
We know that $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$$
\begin{aligned}
& \frac{3}{5}=\frac{3}{10}+\frac{2}{5}-\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{10} \\
& \therefore \text { Given expression }=\frac{\left(\frac{1}{10}\right)}{\left(\frac{2}{5}\right)} \times \frac{\left(\frac{1}{10}\right)}{\left(\frac{3}{10}\right)}=\frac{1}{4} \times \frac{1}{3}=\frac{1}{12}
\end{aligned}
$$
\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right) \times \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})} \times \frac{\mathrm{P}(\mathrm{B} \cap \mathrm{A})}{\mathrm{P}(\mathrm{A})}
$$
We know that $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$
$$
\begin{aligned}
& \frac{3}{5}=\frac{3}{10}+\frac{2}{5}-\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{10} \\
& \therefore \text { Given expression }=\frac{\left(\frac{1}{10}\right)}{\left(\frac{2}{5}\right)} \times \frac{\left(\frac{1}{10}\right)}{\left(\frac{3}{10}\right)}=\frac{1}{4} \times \frac{1}{3}=\frac{1}{12}
\end{aligned}
$$
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