Question

If $A$ and $B$ are two independent events such that $P\left(A\right)>0.5,P\left(B\right)>0.5,$ $P(A\cap \overline{B})=\frac{3}{25}$, $P(\overline{A}\cap B)=\frac{8}{25}$, then $P(A\cap B)=k$, then $\frac{25k}{2}$ is

Answer 1

$\frac{3}{25}=P\left(A\right)-P(A\cap B)=P\left(A\right)-P\left(A\right)P\left(B\right)$

$\frac{8}{25}=P\left(B\right)-P(A\cap B)=P\left(B\right)-P\left(A\right).P\left(B\right)$

$P\left(B\right)-P\left(A\right)=\frac{1}{5}$

Let $P\left(A\right)=a,P\left(B\right)=b$
$\Rightarrow a-ab=\frac{3}{25},b-ab=\frac{8}{25},b-a=\frac{1}{5}$

$\therefore a-a(a+\frac{1}{5})=\frac{3}{25}$

$25a^2-20a+3=0$

$a=\frac{3}{5}$, $b=\frac{3}{5}+\frac{1}{5}=\frac{4}{5}$

$P(A\cap B)=\frac{4}{5}.\frac{3}{5}=\frac{12}{25}$