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 the value of $P(A\cap B)$ is

• A. $\frac{12}{25}$
• B. $\frac{14}{25}$
• C. $\frac{18}{25}$
• D. $\frac{24}{25}$

Answer 1

The correct option is A $\frac{12}{25}$

Given : $P(A\cap \overline{B})=\frac{3}{25},P(\overline{A}\cap B)=\frac{8}{25}$
As $A$ and $B$ are independent events, so
$P(A\cap B)=P\left(A\right)\times P\left(B\right)$
Assuming $P\left(A\right)=a,P\left(B\right)=b$
$P\left(A\right)-P(A\cap B)=\frac{3}{25}$
$\Rightarrow P\left(A\right)-P\left(A\right)\times P\left(B\right)=\frac{3}{25}\Rightarrow a-ab=\frac{3}{25}\cdots \left(1\right)P\left(B\right)-P(A\cap B)=\frac{8}{25}\Rightarrow P\left(B\right)-P\left(A\right)\times P\left(B\right)=\frac{8}{25}\Rightarrow b-ab=\frac{8}{25}\cdots \left(2\right)$

From equations $\left(1\right)$ and $\left(2\right),$ we get
$a(1-\frac{8}{25(1-a)})=\frac{3}{25}\Rightarrow a(17-25a)=3(1-a)\Rightarrow 25a^2-20a+3=0\Rightarrow 25a^2-15a-5a+3=0\Rightarrow (5a-1)(5a-3)=0\Rightarrow a=\frac{3}{5}(\because P(A)>0.5)$
From equation $\left(2\right),$ we get
$b=\frac{4}{5}\therefore P(A\cap B)=P\left(A\right)\times P\left(B\right)=\frac{12}{25}$