A factor has three machines A, B and C, which produce 100,200, and 300 items of a particular type daily. The machines produce 2%, 3% and 5% defective items respectively. One day when the production was over, an item was picked up randomly and it was found to be defective. Find the probability that it was produced by machine A.

Explanation

Given that ,
A factor has three machines `A, B and C`, which produce `100,200, and 300`
items of a particular type daily.
The machines produce` 2%, 3% and 5%`
defective items respectively.
Let `E_1,E_2,E_3` is machine `A,B,C`
`D=text(defective items)`

total production is`100+200+300=600`
`p(E_1)=100/600=1/6`
`p(E_2)=200/600=1/3`
`p(E_3)=300/600=1/2`
now, `p(D/E_1)=0.02`
`p(D/E_2)=0.03`
`p(D/E_3)=0.05`
`p(E_1/D)=(p(E_1)*p(D/E_1))/(p(E_2)*p(D/E_2)+p(E_2)*p(D/E_3)+p(E_3)*p(D/E_3))`
`=(1/6xx0.02)/(1/6xx0.02+1/3xx0.03+1/2xx0.05)`
`=2/23`