A factory has two machines and . Past record shows that machine A produced of the items of output and machine B produced of the items. Further, of the items produced by machine and produced by machine were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?
A factory has two machines and . Past record shows that machine A produced of the items of output and machine B produced of the items. Further, of the items produced by machine and produced by machine were defective. All the items are put into one stockpile and then one item is chosen at random from this and is found to be defective. What is the probability that it was produced by machine B?

Solution:

Let represent the event that item A produces, represent the event that item produces, and represent the event that the generated product is discovered to be defective.

Then  Also (item is defective given that it is produced by machine  Likewise, (thing is defective because it was made by a machine $left.B\right)=1 \%=1 / 100 \mathrm{B} \mathrm{P}\left(\mathrm{E}_{2} \mid \mathrm{A}\right) \mathrm{P}\left(\mathrm{E}_{2} \mid \mathrm{A}\right)=\frac{\mathrm{P}\left(\mathrm{E}_{2}\right) \cdot \mathrm{P}\left(\mathrm{X} \mid \mathrm{E}_{2}\right)}{\mathrm{P}\left(\mathrm{E}_{1}\right) \cdot \mathrm{P}\left(\mathrm{X} \mid \mathrm{E}_{1}\right)+\mathrm{P}\left(\mathrm{E}_{2}\right) \cdot \mathrm{P}\left(\mathrm{X} \mid \mathrm{E}_{2}\right)} =\frac{\frac{2}{5} \times \frac{1}{100}}{\frac{3}{5} \times \frac{2}{100}+\frac{2}{5} \times \frac{1}{100}} =\frac{\frac{2}{5} \times \frac{1}{100}}{\frac{1}{500}(6+2)}=\frac{2}{8}\Rightarrow \mathrm{P}\left(\mathrm{E}_{2} \mid \mathrm{A}\right)=\frac{1}{4}$