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Probabilities of solving a specific problem independently by A and B are `1/2 a n d1/3` 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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Let `P(A)` is the probability that `A` solves the problem and `P(B)` is the probability that `B` solves the problem.
Here, `P(A) = 1/2 and P(B) = 1/3`
`:. P(barA) = 1-1/2 = 1/2`
`P(barB) = 1-1/3 = 2/3`
(i) Probability when the problem is solved:
In this case either one of them or both of them can solve the problem.
So, required probability,
`=P(A)P(barB)+P(B)P(barA)+P(A)P(B)`
`=1/2*2/3+1/3*1/2+1/2*1/3 = 1/3+1/6+1/6 `
`=2/3.`

(ii) Probability when exactly one of them solve the problem :
In this case, required probability `= P(A)P(barB)+P(B)P(barA)`
`=1/2*2/3+1/3*1/2 = 1/3+1/6`
`=1/2.`

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