Let A and B be independent events such that P ( A ) = p, P ( B ) =...

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

$P\left(A \cap \overset{¯}{B}\right) + P\left(\overset{¯}{A} \cap B\right) = \dfrac{5}{9}$

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

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

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

$⇒ P ( A ) + P ( B ) - 2P ( A ) P ( B ) = \dfrac{5}{9}$

$⇒ p + 2p - 4{p}^{2} = \dfrac{5}{9}$

$⇒ 36{p}^{2} - 27p + 5 = 0$

$⇒ ( 12p - 5 ) ( 3p - 1 ) = 0$

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