The certain item is manufactured by three factories, say 1, 2 and 3. It is known that 1 turns out twice as many items as 2 and that 2 and 3 turn out the same number of items (during a specified production period). It is also known that 2 percent of the items produced by 1 and 2 are defective while 4 percent of those manufactured by 3 are defective. All the items produced are put into one stock pile and then one item is chosen at random. The chosen item was found defective. What is the probability that it was produced in factory 1?
We are given a situation where three factories (1, 2, and 3) are producing an item, and there are certain probabilities of defectiveness associated with each factory's production. The objective is to find the probability that a randomly chosen defective item was produced in factory 1.
Step 1: Define the events and known values
Let:
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P(F1)P(F_1), P(F2)P(F_2), and P(F3)P(F_3) represent the probabilities that the chosen item was produced in factories 1, 2, and 3, respectively.
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The probability that an item from factory 1 is defective is 2%, or P(D∣F1)=0.02P(D|F_1) = 0.02.
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| x /y | −1 | 0 | 1 |
| −1 | α | β | α |
| 0 | β | 0 | β |
| 1 | α | β | α |
| 1 | 2 | 3 | 4 | |
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