Question

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

Answer 1

Let the events be described as below :

$A:$ No change takes place

$E_1:$ Person $A$ gets appointed

$E_2:$ Person $B$ gets appointed

$E_3:$ Person $C$ gets appointed.

The chances of selection of $A,B$ and $C$ are in the ratio $1:2:4$.

Hence,$P\left(E_1\right)=\frac{1}{7},P\left(E_2\right)=\frac{2}{7},P\left(E_3\right)=\frac{4}{7}$

Probabilities of $A,B$ and $C$ introducing changes to improve profits of company are $0.8,0.5$ and $0.3$ respectively. Hence probability of no changes on appointment of $A,B$ and $C$ are $0.2,0.5$ and $0.7$ respectively.

Hence,$P\left(A|E_1\right)=0.2=\frac{2}{10}$

$P\left(A|E_2\right)=0.5=\frac{5}{10}$

$P\left(A|E_3\right)=0.7=\frac{7}{10}$

$\therefore$the required probability is

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

$=\frac{\frac{4}{7}.\frac{7}{10}}{\frac{1}{7}.\frac{2}{10}+\frac{2}{7}.\frac{5}{10}+\frac{4}{7}.\frac{7}{10}}$

$=\frac{\frac{28}{70}}{\frac{2}{70}+\frac{10}{70}+\frac{28}{70}}$

$=\frac{28}{40}=\frac{7}{10}$

$\therefore$ if no change takes place, the probability that it is due to appointment of $C$ is $\frac{7}{10}$