Question

Probability fo 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
Exactly one of them solves the problem

Answer 1

Probability fo solving th eproblem by $A,P\left(A\right)=\frac{1}{2}$

Probability of solving the problem by $B,P,\left(B\right)=\frac{1}{3}$
Probability of not solving the problem by $A=P\left(A^{\prime }\right)=1-P\left(A\right)=1-\frac{1}{2}=\frac{1}{2}$
and probability of not solving the problem by B
$=P\left(B^{\prime }\right)=1-P\left(B\right)=1-\frac{1}{3}=\frac{2}{3}$
P (Exactly one of them solve the problem) = P (A) P(B') + P(A') P(B)
$=\frac{1}{2}\times \frac{2}{3}+\frac{1}{2}\times \frac{1}{3}=\frac{1}{3}+\frac{1}{6}=\frac{2+1}{6}=\frac{3}{6}=\frac{1}{2}$