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Question

A manufacturer has three machine operators A, B an C. The first operatot A produces 1% defective items, where as the other two operators B anc C produce 5% of the time, B on the job for 30% of the time and C on the job for 20% of the time. A defective item is produced, what is the probability that it was produced by A?

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Solution

Let E1 : the event that item is produced by machine A

E2 : the event that item is produced by machine B

and E3 : the event that item is produced by machine C

Here, E1,E2, and E3 are mutually exclusive and exhaustive events.

Moreover,

P(E1)=50%=50100,P(E2)=30%=30100 and P(E3)=20%=20100
Let E : The event that item chosen is found to be defective',

P(EE1)=1100,P(EE2)=5100,andP(EE3)=7100.

By using Baye's therorem, we obtain

P(E1E)=P(EE1)P(E1)P(EE1)P(E1)+P(EE2)P(E2)+P(EE3)P(E3)=1100×501001100×50100+5100×30100+7100×20100=5050+150+140=50340=534


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