Probability of solving specific problem independently by A and B are 1

1.	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 (i) the problem is solved (ii) exactly one of them solves the problem.
Answer» Probability of solving the problem by A, P(A) = 1/2 Probability of solving the problem by B, P(B) = 1/3 Probability of not solving the problem by A = P(A’) = 1 – P(A) = 1 - 1/2 = 1/2 and probability of not solving the problem by B = P(B’) = 1 – P(B) = 1 - 1/3 = 2/3 (i) P (the problem is solved) = 1 – P(none of them solve the problem) = 1 (A' ∩ B') = 1 - P(A')P(B') (since A and B are independent A’ and B’ are independent) = 1 - (1/2 x 2/3) = 1 - 1/3 = 2/3 (ii) P (exactly one of them solve the problem) = P(A) P(B’) + P(A’) P(B) = 1/2 x 2/3 + 1/2 x 1/3 = 1/3 + 1/6 = (2 + 1)/6 = 3/6 = 1/2

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

Answer»

Probability of solving the problem by A, P(A) = 1/2

Probability of solving the problem by B, P(B) = 1/3

Probability of not solving the problem by A = P(A’) = 1 – P(A) = 1 - 1/2 = 1/2

and probability of not solving the problem by B = P(B’) = 1 – P(B) = 1 - 1/3 = 2/3

(i) P (the problem is solved) = 1 – P(none of them solve the problem)

= 1 (A' ∩ B') = 1 - P(A')P(B')

(since A and B are independent A’ and B’ are independent)

= 1 - (1/2 x 2/3) = 1 - 1/3 = 2/3

(ii) P (exactly one of them solve the problem) = P(A) P(B’) + P(A’) P(B)

= 1/2 x 2/3 + 1/2 x 1/3 = 1/3 + 1/6 = (2 + 1)/6 = 3/6 = 1/2

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