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प्रश्न
A manufacturer has three machine operators A, B and C. The first operator A produces 1% defective items, where as the other two operators B and C produce 5% and 7% defective items respectively. A is on the job for 50% of the time, B is on the job for 30% of the time and C is on the job for 20% of the time. A defective item is produced, what is the probability that was produced by A?
उत्तर
Let E1, E2, and E3 be the event that the machine is operated by A, B, and C, respectively.
Let A be the event of producing defective items.
∴ `P (E_1) = 50% = 1/2`
`P (E_2) = 30% = 3/10`
`P (E_3) = 20% = 1/5`
Now,
`P(A/E_1) = 1% = 1/100`
`P(A/E_2) = 5% = 5/100`
`P(A/E"_2) = 7% = 7/100`
Using Bayes' theorem, we get
Required probability = `P (E_1/A) = (P(E_1)P(A/E_1))/(P(E_1)P(A/E_1)+P(E_2)P(A/E_2)+P(E_2)P(A/E_2)`
= `(1/2 xx 1/100)/(1/2 xx 1/100 + 3/10 xx 5/100 + 1/5 xx 7/100)`
= `(1/2)/(1/2 + 15/10 + 7/5)`
= `(1/2)/((5 + 15 + 14)/10)`
= `5/34`
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