Probability (exactly one of $\mathrm{A},\mathrm{B}$ occurs) $=\frac{5}{9}$

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

$P\left(A\right)-P\left(A\cap B\right)+P\left(B\right)-P\left(A\cap B\right)=\frac{5}{9}$

Given, $A$ and $B$ are independent events.

So, $P\left(A\cap B\right)=P\left(A\right)P\left(B\right)$

$\Rightarrow P\left(A\right)+P\left(B\right)-2P\left(A\right)P\left(B\right)=\frac{5}{9}$

$\Rightarrow p+2p-4p^2=\frac{5}{9}$

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

$\Rightarrow (12p-5)(3p-1)=0$

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