In a factory, machine A produces $\displaystyle 30$% of total output, machine B produces $\displaystyle 25$% and the machine C produces the remaining output. The defective items produced by machines A, B and C are $\displaystyle 1$%, $\displaystyle 1.2$%, $\displaystyle 2$% respectively. An item is picked at random from a day's output and found to be defective. Find the probability that it was produced by machine B? Case Study Based Question

\[\begin{aligned} & E_{1}: \text { Item was produced by } A \\ & E_{2}: \text { Item was produced by } B \\ & E_{3}: \text { Item was produce by } C \\ & F: \text { Item was defective } \\ & P\left(E_{1}\right)=\frac{30}{100}, P\left(E_{2}\right)=\frac{25}{100}, P\left(E_{3}\right)=\frac{45}{100} \\ & P\left(F \mid E_{1}\right)=\frac{1}{100}, P\left(F \mid E_{2}\right)=\frac{1 \cdot 2}{100}, P\left(F \mid E_{3}\right)=\frac{2}{100} \\ & 2 \mid F)=\frac{P\left(E_{2}\right) P\left(F \mid E_{2}\right)}{P\left(E_{1}\right) P\left(F \mid E_{1}\right)+P\left(E_{2}\right) P\left(F \mid E_{2}\right)+P\left(E_{3}\right) P\left(F \mid E_{3}\right)} \\ & =\frac{\dfrac{25}{100} \times \dfrac{1 \cdot 2}{100}}{\dfrac{30}{100} \times \dfrac{1}{100}+\dfrac{25}{100} \times \dfrac{1.2}{100}+\dfrac{45}{100} \times \dfrac{2}{100}} \\ & =\frac{3}{3+3+9}=\frac{1}{5} \end{aligned} \]