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
(i) the problem is solved
(ii) exactly one of them solves the problem.

Answer 1

Solution:

Probability that A solves the problem p(A) = 1/2

Probability that A does not solves the problem p(A bar) = 1/2

Probability that B solves the problem p(B) = 1/3

Probability that B does not solves the problem p(B bar) = 2/3

find the probability that
(i) the problem is solved :
There are three cases for problem is to be solved

a) Both A and B solve the problem= p(A)p(B)

b) A solved but B didn't:p(A)p(B bar)

c) B solved but A didn't:p(A bar)p(B)

Probability that problem is solved
$= \frac{1}{2} \times \frac{1}{3} + \frac{1}{2} \times \frac{2}{3} + \frac{1}{2} \times \frac{1}{3} \\ \\ = \frac{1}{6} + \frac{2}{6} + \frac{1}{6} \\ \\ = \frac{4}{6} \\ \\ = \frac{2}{3}$


(ii) exactly one of them solves the problem:

a) A solved but B didn't:p(A)p(B bar)

b) B solved but A didn't:p(A bar)p(B)

Probability that exactly one of them solve the problem
$= \frac{1}{2} \times \frac{1}{3} + \frac{1}{2} \times \frac{2}{3} \\ \\ = \frac{1}{6} + \frac{2}{6} \\ \\ = \frac{3}{6} \\ \\ = \frac{1}{2}$
Hope it helps you.