Question

In a factory manufacturing bolts; machines $A,B$ and $C$ manufacture respectively $20\%,30\%$ and $50\%$ of the total production. Of their outputs, $2\%,3\%$ and $5\%$ are defective bolts. A bolt is drawn at random from the product and is found to be defective. What is the probability that it was manufactured by machine $C$?

Answer 1

Let $A,B,C$ be the events that a bolt selected at random is manufactured respectively by machines $A,B$ and $C$ respectively so that $P\left(A\right)=\frac{2}{10};P\left(B\right)=\frac{3}{10}$ and $P\left(C\right)=\frac{5}{10}$.

Let $D$ be the event that the bolt selected at random is defective, so that

$P\left(D/A\right)=\frac{2}{100}$

$P\left(D/B\right)=\frac{3}{100}$

And $P\left(D/C\right)=\frac{5}{100}$

$P\left(C/D\right)=\frac{P\left(D/C\right).P\left(C\right)}{P\left(D/A\right)P\left(A\right)+P\left(D/B\right)P\left(B\right)+P\left(D/C\right)P\left(C\right)}$

$=\frac{5\times 5}{4+9+25}=\frac{25}{38}$