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 occur together is :

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

Answer 1

The correct option is A $0.10$

$P\left(\text{A or B}\right)=P(A\cup B)=\frac{1}{2}$
$P\left(\text{exactly one of A or B}\right)=\frac{2}{5}$
$\Rightarrow P\left(A\right)-P(A\cap B)+P\left(B\right)-P(A\cap B)=\frac{2}{5}$
$\Rightarrow P(A\cup B)-P(A\cap B)=\frac{2}{5}$
$\Rightarrow P(A\cap B)=\frac{1}{10}$