Question

The odds against solving a problem by \(\mathrm{A}\) and \(\mathrm{B}\) are \(3: 2\) and \(2: 1\) respectively, then the probability that the problem will be solved, is

Answer 1

Given,
\[
\mathrm{P}(\mathrm{A})=\frac{3}{2+3}=\frac{3}{5}, \mathrm{P}(\mathrm{B})=\frac{2}{2+1}=\frac{2}{3}
\]
and
\[
\mathrm{P}(\overline{\mathrm{A}})=1-\frac{3}{5}=\frac{2}{5}, \mathrm{P}(\overline{\mathrm{B}})=1-\frac{2}{3}=\frac{1}{3}
\]
\(\therefore\) Required probability
 

\[=P(A \cap \bar{B})+P(\bar{A} \cap B)+P(\bar{A} \cap \bar{B})\]

\[=P(A) \cdot P(\bar{B})+P(\bar{A}) \cdot P(B)+P(\bar{A}) \cdot P(\bar{A})\]

\[
=\frac{3+4+2}{15}=\frac{9}{15}=\frac{3}{5}
\]