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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 $\mathrm{P}$ (exactly one of $\mathrm{A}, \mathrm{B}$ occurs $)=\frac{5}{9}$, is :
$\frac{1}{3}$
$\frac{2}{9}$
$\frac{4}{9}$
$\frac{5}{12}$
Solution
$\mathrm{P}($ Exactly one of $\mathrm{A}$ or $\mathrm{B}$ )
$=\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})+\mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B})=\frac{5}{9}$
$=\mathrm{P}(\mathrm{A}) \mathrm{P}(\overline{\mathrm{B}})+\mathrm{P}(\overline{\mathrm{A}}) \mathrm{P}(\mathrm{B})=\frac{5}{9}$
$\Rightarrow \mathrm{P}(\mathrm{A})(1-\mathrm{P}(\mathrm{B}))+(1-\mathrm{P}(\mathrm{A})) \mathrm{P}(\mathrm{B})=\frac{5}{9}$
$\Rightarrow \mathrm{p}(1-2 \mathrm{p})+(1-\mathrm{p}) 2 \mathrm{p}=\frac{5}{9}$
$\Rightarrow 36 \mathrm{p}^{2}-27 \mathrm{p}+5=0$
$\Rightarrow \mathrm{p}=\frac{1}{3} \text { or } \frac{5}{12}$
$\mathrm{p}_{\max }=\frac{5}{12}$
