A manufacturer has three machine operators A, B and C. The first operator A produces 1\% defective items, where as the other two operators B and C produce 5\% and 7\% defective items respectively. A is on the job for 50\% of the time, B is on the job for 30\% of the time and C is on the job for 20\% of the time. A defective item is produced, what is the probability that it was produced by A?

.: Let ${E_1}$ : `Item is produced by machine $A$
,
${E_2}$ : Item is produced by machine $B$ ’

and ${E_3}$ : `ltem is produced by machine $C$
’,
$\Rightarrow P\left( {{E_1}} \right) = \cfrac{{50}}{{100}},P\left( {{E_2}} \right) = \cfrac{{30}}{{100}}$

and $P\left( {{E_3}} \right) = \cfrac{{20}}{{100}}$

Let A Item chosen is found to be defective',

Then $P\left( {A|{E_1}} \right) = \cfrac{1}{{100}},P\left( {A|{E_2}} \right) = \cfrac{5}{{100}}$

and $P\left( {A|{E_3}} \right) = \cfrac{7}{{100}}$

Hence, required probability is

$P\left( {{E_1}|A} \right) = \cfrac{{P\left( {A|{E_1}} \right)P\left( {{E_1}} \right)}}{{P\left( {A|{E_1}} \right)P\left( {{E_1}} \right) + P\left( {A|{E_2}} \right)P\left( {{E_2}} \right) + P\left( {A|{E_3}} \right)P\left( {{E_3}} \right)}}$

$= \cfrac{{\cfrac{1}{{100}} \times \cfrac{{50}}{{100}}}}{{\cfrac{1}{{100}} \times \cfrac{{50}}{{100}} + \cfrac{5}{{100}} \times \cfrac{{30}}{{100}} + \cfrac{7}{{100}} \times \cfrac{{20}}{{100}}}} = \cfrac{{50}}{{50 + 150 + 140}} = \cfrac{{50}}{{340}} = \cfrac{5}{{34}}$

.