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.

Given:

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

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

Because A and B both are independent.

⇒ P (A ∩ B) = P(A) . P(B)

⇒ P (A ∩ B) =

P(A^{’}) = 1 – P(A) = 1 – 1/2 = 1/2

P(B^{’}) = 1 – P(B) =

(i) the problem is solved

The problem is solved, i.e. it is either solved by A or it is solved by B.

= P(A ∪ B)

As we know, P (A ∪ B) = P(A) + P(B) - P (A ∩ B)

⇒ P (A ∪ B) =

(ii) exactly one of them solves the problem

i.e. either problem is solved by A but not by B or vice versa

i.e. P(A).P(B^{’}) + P(A^{’}).P(B)

=

=

⇒ P(A).P(B^{’}) + P(A^{’}).P(B) = 1/2

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