Question

If $A$ and $B$ are independent event such that $P(A\cap B^{\prime })=\frac{3}{25}$ and $P(A^{\prime }\cap B)=\frac{8}{25}$, then $P\left(A\right)=$

• A. $1/5$
• B. $3/8$
• C. $2/5$
• D. $4/5$

Answer 1

The correct option is C $2/5$

$A$ & $B$ are independent

$P(A\cap B)=\frac{3}{25}$ and $P(A^{\prime }\cap B)=\frac{8}{25}P\left(A\right)=(?)$

$\to P\left(A\right)+P\left(B\right)=1---\left(i\right)$ ($A$& $B$ are independent )

$\to P(A\cap B^{\prime })=P\left(A\right)-P(A\cap B)$

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

$\to P(A^{\prime }\cap B)=P\left(B\right)-P(A\cap B)$

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

$\to$ solving equation $\left(ii\right)$ and $\left(iii\right)$

$\frac{P\left(A\right)-P(A\cap B)=\frac{3}{25}P\left(B\right)-P(A\cap B)=\frac{8}{25}-+-}{P\left(A\right)-P\left(B\right)=\frac{3}{25}-\frac{8}{25}P\left(A\right)-P\left(B\right)=\frac{3-8}{25}}$

$=\frac{-5}{25}$

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

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

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

$\therefore 2P\left(A\right)=\frac{-1}{5}+1$

$\therefore 2P\left(A\right)=\frac{4}{5}$

$\therefore P\left(A\right)=\frac{4}{5\times 2}$

$\therefore P\left(A\right)=\frac{2}{5}$