$\mathrm{P}($ Exactly one of $\mathrm{A}$ or $\mathrm{B}$ )

$=\mathrm{P}(\mathrm{A} \cap \overline{\mathrm{B}})+\mathrm{P}(\overline{\mathrm{A}} \cap \mathrm{B})=\frac{5}{9}$

$=\mathrm{P}(\mathrm{A}) \mathrm{P}(\overline{\mathrm{B}})+\mathrm{P}(\overline{\mathrm{A}}) \mathrm{P}(\mathrm{B})=\frac{5}{9}$

$\Rightarrow \mathrm{P}(\mathrm{A})(1-\mathrm{P}(\mathrm{B}))+(1-\mathrm{P}(\mathrm{A})) \mathrm{P}(\mathrm{B})=\frac{5}{9}$

$\Rightarrow \mathrm{p}(1-2 \mathrm{p})+(1-\mathrm{p}) 2 \mathrm{p}=\frac{5}{9}$

$\Rightarrow 36 \mathrm{p}^{2}-27 \mathrm{p}+5=0$

$\Rightarrow \mathrm{p}=\frac{1}{3} \text { or } \frac{5}{12}$

$\mathrm{p}_{\max }=\frac{5}{12}$