Question

Probability of solving specific problem independently by $A$ and $B$ are $\frac{1}{2}$ and $\frac{1}{3}$ respectively. If both try to solve the problem independently, find the probability that
exactly one of them solves the problem.

Answer 1

Exactly one solve

$=P\left(\overline{A}B\right)+P\left(A\overline{B}\right)$

$=P\left(\overline{A}\right)P\left(B\right)+P\left(A\right)P\left(\overline{B}\right)$

$=(1-\frac{1}{2})\times \frac{1}{3}+\frac{1}{2}\times (1-\frac{1}{3})$

$=\frac{1}{2}\times \frac{1}{3}+\frac{1}{2}\times \frac{2}{3}=\frac{3}{6}=\frac{1}{2}$