Question

A factory has three machines, A, B, and C, for producing items. Machine A produces 50%, B produces 30%, and C produces 20%. If a randomly selected item from the factory's output is found to be defective, what is the probability that B made it?

It is also known that 3% of the items produced by machine A are defective, as are 2% of the items produced by machine B and 1% of the items from machine C.

If a randomly selected item from the factory's output is found to be defective, what is the probability that it was produced by machine B? Also, if a randomly selected chip is found not to be defective, what is the probability that it came from machine B?

Answer 1

Given the item is defective, probability of it being produced by B`=0.26`

Explanation:

Prior information

Probability of an item produced by Machine A `P(A)=50%=0.5`
Probability of an item produced by Machine B `P(B)=30%=0.3`
Probability of an item produced by Machine C `P(C)=20%=0.2`

On the basis of additional information
Given the item produced by A, probability of it being defective `P(D/A)=3% = 0.03`
Given the item produced by B, probability of it being defective `P(D/B)=2% = 0.02`
Given the item produced by C, probability of it being defective `P(D/C)=1% = 0.01`

Given the item is defective, probability of it being produced by B

`P(B/D)=[P(B)xxP(D/B)] /[[P(B) xx P(D/B)]+[P(A) xx P(D/A)]+[P(C) xx P(D/C)]`

`=[0.3xx0.02]/[[0.3xx0.02]+[0.5xx0.03]+[0.2xx0.01]`
`0.006/[0.006+0.015+0.002]=0.006/0.023=0.26`

Given the item is defective, probability of it being produced by B`=0.26`