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Question
Probability of solving specific problem independently by A and B are `1/2` and `1/3` respectively. If both try to solve the problem independently, find the probability that
- the problem is solved
- exactly one of them solves the problem.
Solution
According to the question, P(A) = `1/2`, P(B) = `1/3`
∴ `"P"(overline"A") = 1 - "P"("A") = 1 - 1/2 = 1/2`
and `"P"(overline"B") = 1 - "P"("B") = 1 - 1/3 = 2/3`
i. ∴ Probability that the problem is not solved by both = `"P"(overlineA ∩ overlineB) = P(overlineA) . P(overlineB)`
= `1/2 xx 2/3`
= `1/3`
∴ The probability that at least one solves the problem
= `1 - P(overlineA ∩ overlineB)`
= `1 - 1/3`
= `2/3`
ii. The probability that only one person will solve the problem
= P(A ∩ B') + P(A ∩ B')
= P(A) . P(B') + P(A') . P(B)
= `1/2 xx 2/3 + 1/2 xx 1/3`
= `2/6 + 1/6`
= `3/6`
= `1/2`
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