Question

Let $A$ and $B$ be independent events such that $P\left(A\right)=p,P\left(B\right)=2p.$ The largest value of $p$ for which $P\text{(exactly one of}A,B\text{occurs)}=\frac{5}{9},$ is

• A. $\frac{5}{12}$
• B. $\frac{2}{9}$
• C. $\frac{4}{9}$
• D. $\frac{1}{3}$

Answer 1

The correct option is A $\frac{5}{12}$

$P\text{(exactly one of}A,B\text{occurs)}$
$=P(A\cap \overline{B})+P(\overline{A}\cap B)$
$=P\left(A\right)\times P\left(\overline{B}\right)+P\left(\overline{A}\right)\times P\left(B\right)$
$=p(1-2p)+(1-p)2p$
$\Rightarrow 3p-4p^2=\frac{5}{9}$
$\Rightarrow 27p-36p^2=5$
$\Rightarrow 36p^2-27p+5=0$
$\Rightarrow p=\frac{27\pm 3}{72}=\frac{5}{12}\text{and}\frac{1}{3}$
$\therefore p_{max}=\frac{5}{12}$