If $A$ and $B$ are two events such that $P(A)=0.7$, $\mathrm{P}(\mathrm{B})=0.4$ and $\mathrm{P}(\mathrm{A}…

If $A$ and $B$ are two events such that $P(A)=0.7$, $\mathrm{P}(\mathrm{B})=0.4$ and $\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})=0.5$, where $\overline{\mathrm{B}}$ denotes the complement of $B$, then $P(B \mid(A \cup \bar{B}))$ is equal:-
  1. $\frac{1}{4}$
  2. $\frac{1}{2}$
  3. $\frac{1}{6}$
  4. $\frac{1}{3}$

Solution

$\begin{aligned} & \mathrm{P}(\mathrm{A})=\frac{7}{10}, \mathrm{P}(\mathrm{B})=\frac{4}{10} \\ & \mathrm{P}(\mathrm{A} \cup \overline{\mathrm{B}})=\frac{5}{10} \\ & \mathrm{P}\left(\frac{\mathrm{B}}{\mathrm{A} \cup \overline{\mathrm{B}}}\right)=\frac{\mathrm{P}(\mathrm{B} \cap(\mathrm{A} \cup \overline{\mathrm{B}}))}{\mathrm{P}(\mathrm{A} \cup \overline{\mathrm{B}})} \\ & =\frac{\mathrm{P}((\mathrm{B} \cap \overline{\mathrm{B}}) \cup(\mathrm{B} \cap \mathrm{A}))}{\mathrm{P}(\mathrm{A} \cup \overline{\mathrm{B}})}=\frac{\mathrm{P}(\mathrm{A} \cap \mathrm{B})}{\mathrm{P}(\mathrm{A} \cup \overline{\mathrm{B}})}\end{aligned}$
$\begin{aligned} & =\frac{\mathrm{P}(\mathrm{A})-\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})}{\mathrm{P}(\mathrm{A})+\mathrm{P}(\overline{\mathrm{B}})-\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})}=\frac{\frac{7}{10}-\frac{5}{10}}{\frac{7}{10}+\left(1-\frac{4}{10}\right)-\frac{5}{10}} \\ & =\frac{2}{8}=\frac{1}{4}\end{aligned}$ /

Asked in: JEE Main 2025 (08 Apr Shift 2)

Practice more Probability questions on Aicharya