Question

The odds against solving a problem by $A$ and $B$ are $3:2$ and $2:1$ respectively, then the probability that the problem will be solved, is

• A. $\frac{3}{5}$
• B. $\frac{2}{15}$
• C. $\frac{2}{5}$
• D. $\frac{11}{15}$

Answer 1

The correct option is A $\frac{3}{5}$

Given,

$P\left(A\right)=\frac{3}{2+3}=\frac{3}{5},P\left(B\right)=\frac{2}{2+1}=\frac{2}{3}$
and

$P\left(\overline{A}\right)=1-\frac{3}{5}=\frac{2}{5},P\left(\overline{B}\right)=1-\frac{2}{3}=\frac{1}{3}$

$\therefore$ Required probability
$=P(A\cap \overline{B})+P(\overline{A}\cap B)+P(\overline{A}\cap \overline{B})$$=P\left(A\right)\cdot P\left(\overline{B}\right)+P\left(\overline{A}\right)\cdot P\left(B\right)+P\left(\overline{A}\right)\cdot P\left(\overline{A}\right)$

$=\frac{3}{5}\cdot \frac{1}{3}+\frac{2}{5}\cdot \frac{2}{3}+\frac{2}{5}\cdot \frac{1}{3}$

$=\frac{1}{5}+\frac{4}{15}+\frac{2}{15}$

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