# 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?

Feb 11, 2018

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]
$\frac{0.006}{0.006 + 0.015 + 0.002} = \frac{0.006}{0.023} = 0.26$

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