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 $\frac{1}{2}$