Question

The probability of solving the specific problem independently by A and B are $\frac{1}{2}and\frac{1}{2}$ respectively, If both try to solve the problem independently, find the probability that
(i) the problem is solved
(ii) exactly one of them solves the problem.

Answer 1

$1).1-P\left(\overline{A}\right).P\left(\overline{B}\right)$

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

$=1-\frac{1}{2}\times \frac{1}{2}$

$=1-\frac{1}{4}$

$=\frac{3}{4}.$

$2).P\left(A\right)\times P\left(\overline{B}\right)+P\left(B\right)\times P\left(\overline{A}\right)$

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

$=\frac{1}{4}+\frac{1}{4}$

$=\frac{1}{2}.$

Hence, solved.