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?

1 Answer
Feb 11, 2018

Answer:

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#