Question

If $A$ and $B$ are two independent events with $P\left(A\right)=\frac{1}{2},P(A\cup B)=\frac{3}{5}$ and $P\left(B\right)=x$, then find the value of $x$.

Answer 1

Fact: If A and B are independent events, then $P(A\cap B)=P\left(A\right)P\left(B\right)$

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

$\Rightarrow \frac{3}{5}=\frac{1}{2}+x-P\left(A\right)P\left(B\right)$, substitute the given value

$\Rightarrow \frac{3}{5}=\frac{1}{2}+x-\frac{1}{2}x$

$\Rightarrow \frac{x}{2}=\frac{3}{5}-\frac{1}{2}=\frac{1}{10}$

$\therefore x=\frac{1}{5}$