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

\[\left.\begin{array}{l} E_{1}: A \text { is selected } \\ E_{2}: B \text { is selected } \\ E_{2}: C \text { is selected } \end{array}\right\} \] \[\begin{aligned} & E_{3}: C \text { is selected } \\ & F: \text { increase in profit does not take place } \\ & 1)=1 / 7, P\left(E_{2}\right)=2 / 7, P\left(E_{3}\right)=4 / 7 \\ & \left.\mid E_{1}\right)=0 \cdot 2, P\left(F \mid E_{2}\right)=0 \cdot 5, P\left(F \mid E_{3}\right)=0 \cdot 7 \\ & 1 \mid F)=\frac{P\left(E_{1}\right) P\left(F \mid E_{1}\right)}{P\left(E_{1}\right) P\left(F \mid E_{1}\right)+P\left(E_{2}\right) P\left(F \mid E_{2}\right)+P\left(E_{3}\right) P\left(F \mid E_{3}\right)} \\ & \quad=\frac{\dfrac{1}{7} \times \dfrac{2}{10}}{\dfrac{1}{7} \times \dfrac{2}{10}+\dfrac{2}{7} \times \dfrac{5}{10}+\dfrac{4}{7} \times \dfrac{7}{10}} \\ & \quad=\frac{2}{40}=\frac{1}{20} \end{aligned} \]