Question

If $A$ and $B$ are two independent events such that $P\left(A\right)=\frac{3}{10}$ and $P(A\cup B)=\frac{4}{5}$, then $P(A\cap B)$ is equal to:

• A. $\frac{1}{5}$
• B. $\frac{1}{10}$
• C. $\frac{3}{14}$
• D. $\frac{3}{35}$

Answer 1

Answer: (C)

$\frac{3}{14}$

$A$ and $B$ are independent event.

$\Rightarrow P(A\cap B)=P\left(A\right).P\left(B\right)$

Now, $P(A\cup B)=P\left(A\right)+P\left(B\right)-P(A\cap B)$

$\Rightarrow \frac{4}{5}=\frac{3}{10}+P\left(B\right)-\left(\frac{3}{10}\right)P\left(B\right)$

$\Rightarrow \frac{4}{5}-\frac{3}{10}=\left(\frac{7}{10}\right)P\left(B\right)$

$\Rightarrow \frac{5}{10}=\left(\frac{7}{10}\right)P\left(B\right)$

$\Rightarrow P\left(B\right)=\frac{5}{7}$

Therefore, $P(A\cap B)=\frac{3}{10}.\frac{5}{7}=\frac{3}{14}$