Question

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

Answer 1

Probability of solving the problem by \(A,P(A)=\frac{1}{2},\)
Probability of solving the problem by \(B,P(B)=\frac{1}{3},\)
Since the problem is solved independently by \(A\) and \(B\),
\(\therefore P(AB)=P(A).P(B)=\frac{1}{2}\times \frac{1}{3}=\frac{1}{6}\)
\(P(A')=1-P(A)=1-\frac{1}{2}=\frac{1}{2}\)
\(P(B')=1-P(B)=1-\frac{1}{3}=\frac{2}{3}\)

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(i)Probability that the problem is solved \(= P (A ∪ B)\)
\(= P (A) + P (B) − P (AB) \)
\(=\frac{1}{2}+\frac{1}{3}-\frac{1}{6}\)
\(=\frac{4}{6}\)
\(=\frac{2}{3}\)

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(ii)Probability that exactly one of them solves the problem is given by,
\(P(A).P(B')+P(B).P(A')\)
\(=\frac{1}{2}×\frac{2}{3}+\frac{1}{2}×\frac{1}{3}\)
\(=\frac{1}{3}+\frac{1}{6}\)
\(=\frac{1}{2}\)