Question

Three persons $A,B$ and $C$ apply for the job of Manager in a Private Company. Chances of their selection $(A,B$ and $C)$ are in the ratio $1:2:4$. The probabilities that $A,B$ and $C$ can introduce changes to improve profits of the company are $0.8,0.5$ and $0.3$ respectively. If the change does not take place, find the probability that it is due to the appointment of $C$

Answer 1

Let $E_1,E_2$ and $E_3$ be the events denoting the selection of $A,B$ and $C$ respectively.

$P\left(E_1\right)=$Probability of selection of $A=\frac{1}{7}$

$P\left(E_2\right)=$Probability of selection of $B=\frac{2}{7}$

$P\left(E_3\right)=$Probability of selection of $C=\frac{4}{7}$

Let $A$ be the events denoting the change not taking place.

$P\left(A|E_1\right)=$Probability that $A$ does not introduce change$=0.2$

$P\left(A|E_2\right)=$Probability that $B$ does not introduce change$=0.5$

$P\left(A|E_3\right)=$Probability that $C$ does not introduce change$=0.7$

$\therefore$ Required probability is $P\left(E_3|A\right)$

By Bayes' theorem , we have

$P\left(E_3|A\right)=\frac{P\left(E_3\right)P\left(A|E_3\right)}{P\left(E_1\right)P\left(A|E_1\right)+P\left(E_2\right)P\left(A|E_2\right)+P\left(E_3\right)P\left(A|E_3\right)}$

$=\frac{\frac{4}{7}\times 0.7}{\frac{1}{7}\times 0.27+\frac{2}{7}\times 0.57+\frac{4}{7}\times 0.7}$

$=\frac{2.8}{0.2+1+2.8}=\frac{2.8}{4}=0.7$