# 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?

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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$= 0.26$

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