Let A and B be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the…
Let A and B be two events such that the probability that exactly one of them occurs is $\frac{2}{5}$ and the probability that A or B occurs is $\frac{1}{2}$, then the probability of both of them occur together is
0.1
0.2
0.01
0.02
Solution
Given that,
$\mathrm{P}\left[\left(\mathrm{~A} \cap \mathrm{~B}^{\prime}\right) \cup\left(\mathrm{A}^{\prime} \cap \mathrm{B}\right)\right]=\frac{2}{5}...(i)$
and $P(A \cup B)=\frac{1}{2}$...(ii)
From (i), we get
$\mathrm{P}(\mathrm{~A})+\mathrm{P}(\mathrm{~B})-2 \mathrm{P}(\mathrm{~A} \cap \mathrm{~B})=\frac{2}{5}...[From(i)]$
$\begin{array}{ll}\therefore & P(A \cup B)-P(A \cap B)=\frac{2}{5} \\ \therefore & \frac{1}{2}-P(A \cap B)=\frac{2}{5} \\ \therefore & P(A \cap B)=\frac{1}{2}-\frac{2}{5}=\frac{1}{10}=0.1\end{array}$