Question

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 occurring together is

• A. $0.10$
• B. $0.20$
• C. $0.01$
• D. $0.02$

Answer 1

The correct option is A

$0.10$

Explanation for the correct option:

Determine the probability of $A$ and $B$ occurring together, $P\left(A\cap B\right)$:

$P\left(A\text{or}B\right)=P\left(A\cup B\right)=\frac{1}{2}\left[\text{Given}\right]$

$$\begin{gathered} P\left(\text{exactly one of them occurs}\right)=\frac{2}{5} \\ P\left(A\right)-P(A\cap B)+P\left(B\right)-P(A\cap B)=\frac{2}{5} \\ P\left(A\cup B\right)-P(A\cap B)=\frac{2}{5}\left[\because P\left(A\right)-P(A\cap B)+P\left(B\right)=P\left(A\cup B\right)\right] \\ P\left(A\cup B\right)-P(A\cap B)=\frac{2}{5}\left[\because P\left(A\cup B\right)=\frac{1}{2}\right] \\ \frac{1}{2}-P(A\cap B)=\frac{2}{5} \\ P(A\cap B)=-\frac{2}{5}+\frac{1}{2} \\ =\frac{-4+5}{10} \\ =\frac{1}{10} \end{gathered}$$

Therefore, $P\left(A\cap B\right)=\frac{1}{10}=0.1$

Hence, option (A) is the correct answer