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Suppose that the four inspectors at a film factory are supposed to stamp the expiration date on each package of film at the end of the assembly line. John, who stamps 20% of the packages, fails to stamp the expiration date once in every 200 packages; Tom, who stamps 60% of the packages, fails to stamp the expiration date once in every 100 packages; Jeff, who stamps 15% of the packages, fails to stamp the expiration date once in every 90 packages; and Pat, who stamps 5% of the packages, fails to stamp the expiration date once in every 200 packages. If a customer complains that her package of film does not show the expiration date, what is the probability that it was inspected by John?
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Think through the following events:
A: the produce is not marked
B1: the produce is marked by John
B2: the produce is marked by Tom
B3: the produce is marked y Jeff
B4: the produce is marked by Pat
Applying the rule of elimination, the formula would be:
P (A) = P (B1) P (A|B1) + P (B2) P (A|B2) + P (B3) P (A|B3) + P (B4) P (A|B4)
And the following are the probabilities:
P(B1)P(A|B1) = (.20)(1/200) = .001
P(B2)P(A|B2) = (.60)(1/100) = .006
P(B3)P(A|B3) = (.15)(1/90) = .00167
P(B4)P(A|B4) = (.05)(1/200) = .00025
The likelihood that the product is not imprinted is equal to .001 + .006 + .00167 + .00025 = .00892
The probability that it was reviewed by John is = .001/.00892 = .1121