The probabilities of solving a specific probl | vedclass

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(B) Let $P(A) = \frac{1}{2}$ be the probability that $A$ solves the problem.Let $P(B) = \frac{1}{3}$ be the probability that $B$ solves the problem.Th

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$\frac{1}{2}$

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(B) Let $P(A) = \frac{1}{2}$ be the probability that $A$ solves the problem.Let $P(B) = \frac{1}{3}$ be the probability that $B$ solves the problem.Then,$P(A') = 1 - P(A) = 1 - \frac{1}{2} = \frac{1}{2}$ and $P(B') = 1 - P(B) = 1 - \frac{1}{3} = \frac{2}{3}$.The probability that exactly one of them solves the problem is given by $P(A \cap B') + P(B \cap A')$.Since the events are independent,this is equal to $P(A) \cdot P(B') + P(B) \cdot P(A')$.$= (\frac{1}{2} 	imes \frac{2}{3}) + (\frac{1}{3} 	imes \frac{1}{2})$$= \frac{2}{6} + \frac{1}{6} = \frac{3}{6} = \frac{1}{2}$.

The probabilities of solving a 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.

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