Question:
Let $A$ and $B$ be independent events such that $\mathrm{P}(\mathrm{A})=\mathrm{p}, \mathrm{P}(\mathrm{B})=2 \mathrm{p} .$ The largest value of $\mathrm{p}$, for which P (exactly one of A, B occurs) $=\frac{5}{9}$, is :
Correct Option: , 4
Solution:
P(Exactly one of A or {B)
$=\mathrm{P}(\mathrm{A}) \mathrm{P}(\overline{\mathrm{B}})+\mathrm{P}(\overline{\mathrm{A}}) \mathrm{P}(\mathrm{B})=\frac{5}{9}$
$\Rightarrow \mathrm{P}(\mathrm{A})(1-\mathrm{P}(\mathrm{B}))+(1-\mathrm{P}(\mathrm{A})) \mathrm{P}(\mathrm{B})=\frac{5}{9}$
$\Rightarrow \mathrm{p}(1-2 \mathrm{p})+(1-\mathrm{p}) 2 \mathrm{p}=\frac{5}{9}$
$\Rightarrow 36 p^{2}-27 p+5=0$
$\Rightarrow \mathrm{p}=\frac{1}{3}$ or $\frac{5}{12}$
$\mathrm{p}_{\max }=\frac{5}{12}$