The probability of solving the problem independently by A and B are given as $\frac{1}{2}$ and $\frac{1}{3}$ respectively.
i.e. P (A) = $\frac{1}{2}$, P (B) = $\frac{1}{3}$,
∴ P (A ∩ B) = P (A) . P (B)
[Since the events corresponding to A and B are independent]
= $\frac{1}{2}\times \frac{1}{3}$ = $\frac{1}{6}$
(i) Probability that the problem is solved
= P (A ∪ B)
= P (A) + P (B) - P (A ∩ B)
= $\frac{1}{2}+\frac{1}{3}-\frac{1}{6}$
= $\frac{3+2-1}{6}$
= $\frac{4}{6}$ = $\frac{2}{3}$
Thus, the probability that the problem is solved is $\frac{2}{3}$
(ii) Probability that exactly one of them solves the problem
= P (A - B) + P (B - A)
= [P (A) - P (A ∩ B) + [P (B) - P (A ∩ B)]
= $\left(\frac{1}{2}-\frac{1}{6}\right)$ + $\left(\frac{1}{3}-\frac{1}{6}\right)$
= $\frac{3-1+2-1}{6}$
= $\frac{3}{5}$ = $\frac{1}{2}$
Thus, the probability that exactly one of them solves the problem is $\frac{1}{2}$