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

(i) $P\left(A\right),P\left(B\right)=\frac{1}{3}$

$P(A\cap B)=P\left(A\right).P\left(B\right)=\frac{1}{6}$

$P\left(Problemissolved\right)=p(A\cup B)=p\left(A\right)+p\left(B\right)-p\left(A\right)\left(B\right)$

$\frac{1}{2}+\frac{1}{3}-\frac{1}{6}$

$\frac{2}{3}$

( ii ) P (exactly one solved problem)

$p\left(Asolve\right)+p\left(Bsoved\right)$

$p(A\cap B)+p(B\cap A)$

$p\left(A\right)-p(A\cap B)+p\left(B\right)-p(B\cap A)$

$\frac{1}{2}+\frac{1}{3}-2\times \frac{1}{6}$

$\frac{1}{2}$