In a bolt factory, machines A , B and C manufacture respectively 20 % ,   30 % and 50 % of the total…

Let $X\equiv$Event that product is defective, then

$$\begin{gathered} P\left(\frac{X}{A}\right)=\frac{3}{100} \\ P\left(\frac{X}{B}\right)=\frac{4}{100} \\ P\left(\frac{X}{C}\right)=\frac{2}{100} \end{gathered}$$

Now,

$P\left(\frac{C}{X}\right)=\frac{P\left(C\right)P\left({\displaystyle \frac{X}{C}}\right)}{P\left(A\right)P\left(\frac{X}{A}\right)+P\left(B\right)P\left(\frac{X}{B}\right)+P\left(C\right)P\left(\frac{X}{C}\right)}$

$\Rightarrow P\left(\frac{C}{X}\right)=\frac{{\displaystyle \frac{50}{100}}\times {\displaystyle \frac{2}{100}}}{\frac{20}{100}\times \frac{3}{100}+\frac{30}{100}\times \frac{4}{100}+\frac{50}{100}\times \frac{2}{100}}$

$\Rightarrow P\left(\frac{C}{X}\right)=\frac{{\displaystyle 100}}{60+120+100}$

$\Rightarrow P\left(\frac{C}{X}\right)=\frac{{\displaystyle 5}}{14}$

Hence this is the correct option.