Question

For A, B and C the chances of being selected as the manager of a firm are in the ratio 4:1:2 respectively. The respective probabilities for them to introduce a radical change in marketing strategy are 0.3, 0.8 and 0.5. If the change does take place, find the probability that it is due to the appointment of B or C.

Answer 1

Let A, E₁, E₂ and E₃ denote the events that the change takes place, A is selected, B is selected and C is selected, respectively.

$$\begin{gathered} \therefore P\left(E_1\right)=\frac{4}{7} \\ P\left(E_2\right)=\frac{1}{7} \\ P\left(E_3\right)=\frac{2}{7} \\ \mathrm{Now}, \\ P\left(A/E_1\right)=0.3 \\ P\left(A/E_2\right)=0.8 \\ P\left(A/E_3\right)=0.5 \\ \mathrm{Using}\mathrm{Bayes}\text{'}\mathrm{theorem},\mathrm{we}\mathrm{get} \\ P\left(E_1/A\right)=\frac{P\left(E_1\right)P\left(A/E_1\right)}{P\left(E_1\right)P\left(A/E_1\right)+P\left(E_2\right)P\left(A/E_2\right)+P\left(E_3\right)P\left(A/E_3\right)} \\ =\frac{{\displaystyle \frac{4}{7}}\times {\displaystyle 0.3}}{{\displaystyle \frac{{\displaystyle 4}}{7}}\times 0.3+{\displaystyle \frac{1}{7}}\times {\displaystyle 0.8+\frac{2}{7}\times 0.5}} \\ =\frac{1.2}{1.2+0.8+1}=\frac{1.2}{3}=\frac{12}{30}=\frac{2}{5} \\ \therefore \mathrm{Required}\mathrm{probability}=1-P\left(E_1/A\right)=1-\frac{2}{5}=\frac{3}{5} \end{gathered}$$