Let A and B be two events such that the
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:

1. (1) $0.02$

2. (2) $0.20$

3. (3) $0.01$

4. (4) $0.10$

Correct Option: , 4

Solution:

$P($ exactly one $)=\frac{2}{5}$

$\Rightarrow \quad P(A)+P(B)-2 P(A \cap B)=\frac{2}{5}$

$P(A$ or $B)=P(A \cup B)=\frac{1}{2}$

$\Rightarrow P(A)+P(B)-P(A \cap B)=\frac{1}{2}$

$\therefore \quad P(A \cap B)=\frac{1}{2}-\frac{2}{5}=\frac{5-4}{10}=\frac{1}{10}=0.10$