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,
$P(A).P(B')+P(B).P(A')$
$=\frac{1}{2}×\frac{2}{3}+\frac{1}{2}×\frac{1}{3}$
$=\frac{1}{3}+\frac{1}{6}$
$=\frac{1}{2}$