Question

Three persons A, B and C apply for a job of 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 changes 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.

Answer 1

\(P(A) = \frac1 {1 + 2 + 4} = \frac17\)

\(P(B) = \frac27 \)

\(P(C)=\frac{4}{7}\)

Let E be the event of increasing in profit.

\(\therefore P\left(\frac EA\right) = 0.8\)

\(P\left(\frac EB\right) = 0.5\)

\(P\left(\frac EC\right) = 0.3\)

\(\therefore P\left(\frac{E'}{A}\right)= 1- 0.8 = 0.2\)

\(P\left(\frac{E'}{B}\right)= 1- 0.5 = 0.5\)

\(P\left(\frac{E'}{C}\right)= 1- 0.3 = 0.7\)

\(P\left(\frac A{E'}\right)= \frac{P\left(\frac{E'}A\right) P(A)}{P\left(\frac{E'}A\right)P(A) + P\left(\frac{E'}{B}\right) P(B) + P\left(\frac{E'}{C}\right)P(C)}\)

\(= \cfrac{0.2 \times \frac17}{0.2 \times \frac17 + 0.5 \times \frac27 + 0.7 \times \frac47}\)

\(= \frac{0.2}{0.2 + 1 + 2.8}\)

\(= \frac{0.2}{4}\)

\(= \frac2{40}\)

\(= \frac1{20}\)

\(= 0.05\)

Hence, the probability that if increases in profit does not take place, then it is due to the appointment of \(A = P\left(\frac A{E_1}\right) = 0.05\)