Three persons $A$, $B$ and $C$ apply for a job of a manager in a private company. Chances of their selection are in the ratio $1 : 2 : 4$. The probability that $A$, $B$ and $C$ can introduce chances to increase the profits of a company are $0.8$, $0.5$ and $0.3$, respectively. If increase in the profit does not take place, find the probability that it is due to the appointment of A.

$\frac{1}{20}$

The correct answer is Option (1) → $\frac{1}{20}$ ##

Let $E_1 = \text{Person A gets the job}$

$E_2 = \text{Person B gets the job}$

$E_3 = \text{Person C gets the job}$

$A = \text{No change takes place}$

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

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

Also, given $P(A/E_1) = 0.2 = \frac{2}{10}, P(A/E_2) = 0.5 = \frac{5}{10}$ and $P(A/E_3) = 0.7 = \frac{7}{10}$

Required probability is:

$ P(E_1/A) = \frac{P(A/E_1)P(E_1)}{P(A/E_1)P(E_1) + P(A/E_2)P(E_2) + P(A/E_3)P(E_3)}$

$= \frac{\frac{2}{10} \times \frac{1}{7}}{\left(\frac{2}{10} \times \frac{1}{7}\right) + \left(\frac{5}{10} \times \frac{2}{7}\right) + \left(\frac{7}{10} \times \frac{4}{7}\right)}$

$= \frac{\frac{2}{70}}{\frac{2}{70} + \frac{10}{70} + \frac{28}{70}} = \frac{2}{40} = \frac{1}{20}$

$∴$ If no change takes place, the probability that it is due to appointment of person A is $\frac{1}{20}$.