Probabilities of solving 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.

Solution

The probability of solving the problem independently by A and B are given as $\frac{1}{2} and \frac{1}{3}$ respectively.

$i . e . P (A) = \frac{1}{2}, P (B) = \frac{1}{3} . ∴ P (A \cap B) = P (A) . P (B)$

[ Since the events corresponding to A and B are independent ]

$= \frac{1}{2} x \frac{1}{3} = \frac{1}{6}$

( i ) Probability that the problem is solved

$= P (A \cup B) = P (A) + P (B) - P (A \cap B) = \frac{1}{2} + \frac{1}{3} - \frac{1}{6} = \frac{3 + 2 - 1}{6} = \frac{4}{6} = \frac{2}{3} Thus, the probability that hte problem is solved is \frac{2}{3} .$

( ii ) Probability that exactly one of them solves the problem

$= P (A - B) + P (B - A) = [P (A) - P (A \cap B) + [P (B) - P (A \cap B)]] = (\frac{1}{2} - \frac{1}{6}) + (\frac{1}{3} - \frac{1}{6}) = \frac{3 - 1 + 2 - 1}{6} = \frac{3}{6} = \frac{1}{2} Thus, the probability that exactly one of them solves the problem is \frac{1}{2} .$

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Probabilities of solving 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.

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Probabilities of solving 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.