If $\mathrm{P}(\mathrm{A})=\frac{3}{10}, \mathrm{P}(\mathrm{B})=\frac{3}{5}, \mathrm{P}(\mathrm{A} \cup…

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 $\mathrm{P}(\mathrm{A} / \mathrm{B}) \times \mathrm{P}(\mathrm{B} / \mathrm{A})=$
  1. $\frac{1}{3}$
  2. $\frac{1}{12}$
  3. $\frac{1}{10}$
  4. $\frac{1}{4}$

Solution

$\mathrm{P}\left(\frac{\mathrm{A}}{\mathrm{B}}\right) \times \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A}}\right)=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{B})} \times \frac{\mathrm{P}(\mathrm{B} \cap \mathrm{A})}{\mathrm{P}(\mathrm{A})}$ We know that $\mathrm{P}(\mathrm{A} \cup \mathrm{B})=\mathrm{P}(\mathrm{A})+\mathrm{P}(\mathrm{B})-\mathrm{P}(\mathrm{A} \cap \mathrm{B})$ $\begin{aligned} & \frac{3}{5}=\frac{3}{10}+\frac{2}{5}-\mathrm{P}(\mathrm{A} \cap \mathrm{B}) \Rightarrow \mathrm{P}(\mathrm{A} \cap \mathrm{B})=\frac{1}{10} \\ & \therefore \text { Given expression }=\frac{\left(\frac{1}{10}\right)}{\left(\frac{2}{5}\right)} \times \frac{\left(\frac{1}{10}\right)}{\left(\frac{3}{10}\right)}=\frac{1}{4} \times \frac{1}{3}=\frac{1}{12} \end{aligned}$

Asked in: MHT CET 2021 (23 Sep Shift 1)

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