Question

If $A$ and $B$ are two independent events such that $P\left(A\right)>\frac{1}{2},P\left(A\cap B^C\right)=\frac{3}{25}$ and $P\left(A^C\cap B\right)=\frac{8}{25},$ then $P\left(A\right)$ is equal to (where, $A^c$ and $B^c$ represent the complement of events $A$ and $B$ respectively)

• A. $\frac{1}{5}$
• B. $\frac{3}{5}$
• C. $\frac{3}{4}$
• D. $\frac{4}{5}$

Answer 1

Answer: (B)

$A,B$ are independent events $P\left(A\cap B^C\right)=P(A)P\left(B^C\right)=P(A)(1-P(B))=\frac{3}{25}\dots$
$P\left(A^C\cap B\right)=P\left(A^C\right)\cdot P(B)=(1-P(A))P(B)=\frac{8}{25}\dots (2)$
From (1) and (2), we get, $P(B)-P(A)=\frac{1}{5}\dots$ (3)
From (1) and (3) $\left(P(B)-\frac{1}{5}\right)(1-P(B))=\frac{3}{25}$
Let, $P(B)=t$ $\Rightarrow \frac{6}{5}t-\frac{1}{5}-t^2=\frac{3}{25}\Rightarrow 30t-5-25t^2=3$
$\Rightarrow 25t^2-30t+8=0\Rightarrow t=\frac{2}{5},\frac{4}{5}$
$P(B)=\frac{2}{5}\Rightarrow P(A)=\frac{1}{5}$
$P(B)=\frac{4}{5}\Rightarrow P(A)=\frac{3}{5}$
$\because P(A)>\frac{1}{2}\Rightarrow P(A)=\frac{3}{5}$