Question

Let $A$ and $B$ be two independent events such that $P\left(A\right)=m$ and $P\left(B\right)=2m.$ If $\alpha ,\beta$ are the values of $m$ for which $P(\text{exactly one of}A,B\text{occurs})=\frac{5}{9},$ then the value of ${\alpha }^2+{\beta }^2$ is equal to

• A. $\frac{51}{144}$
• B. $\frac{31}{144}$
• C. $\frac{29}{144}$
• D. $\frac{41}{144}$

Answer 1

The correct option is D $\frac{41}{144}$

Given, $P(A\cap \overline{B})+P(\overline{A}\cap B)=\frac{5}{9}$
$\Rightarrow P\left(A\right)\cdot P\left(\overline{B}\right)+P\left(\overline{A}\right)\cdot P\left(B\right)=\frac{5}{9}$
$\Rightarrow m(1-2m)+(1-m)2m=\frac{5}{9}$
$\Rightarrow 3m-4m^2=\frac{5}{9}$
$\Rightarrow 36m^2-27m+5=0$
$\Rightarrow m=\frac{5}{12},\frac{1}{3}$
$\therefore {\alpha }^2+{\beta }^2=\frac{25}{144}+\frac{1}{9}=\frac{41}{144}$