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