The chances of $\displaystyle \mathrm{P}, \mathrm{Q}$ and R getting selected as CEO of a company are in the ratio $\displaystyle 4: 1: 2$ respectively. The probabilities for the company to increase its profits from the previous year under the new CEO, P, Q or R are $\displaystyle 0 \cdot 3,0 \cdot 8$ and $\displaystyle 0 \cdot 5$ respectively. If the company increased the profits from the previous year, find the probability that it is due to the appointment of $\displaystyle R$ as CEO.

Let $\displaystyle \mathrm{E}_{1}$ : P is appointed as CEO,$\displaystyle \mathrm{E}_{2}$ : Q is appointed as CEO,$\displaystyle \mathrm{E}_{3}$ : R is appointed as CEOA : company increase profits from previous yearhere, $\displaystyle \mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{4}{7}, \mathrm{P}\left(\mathrm{E}_{3}\right)=\frac{1}{7}, \mathrm{P}\left(\mathrm{E}_{1}\right)=\frac{2}{7}$$\displaystyle \mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{1}\right)=0.3, \mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{2}\right)=0.8, \mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{3}\right)=0.5$$\displaystyle \mathrm{P}\left(\mathrm{E}_{3} \mid \mathrm{A}\right)=\frac{\mathrm{P}\left(\mathrm{E}_{3}\right) \mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{3}\right)}{\mathrm{P}\left(\mathrm{E}_{1}\right) \mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{1}\right)+\mathrm{P}\left(\mathrm{E}_{2}\right) \mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{2}\right)+\mathrm{P}\left(\mathrm{E}_{3}\right) \mathrm{P}\left(\mathrm{A} \mid \mathrm{E}_{3}\right)}$$\displaystyle =\frac{\dfrac{2}{7} \times 0.5}{\dfrac{4}{7} \times 0.3+\dfrac{1}{7} \times 0.8+\dfrac{2}{7} \times 0.5}$$\displaystyle =\frac{1}{3}$