Question

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 30% of the packages, fails to stamp the expiration date once in every 200 packages; Tom, who stamps 50% 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 60 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?

Answer 1

We are given the packages share here as:
P(John) = 0.3,
P(Tom) = 0.5,
P(Jeff) = 0.15,
P(Pat) = 0.05

Also, we are given the conditional probabilities for stamp failure is computed here as:
P(fail to stamp | John) = 1/200 = 0.005
P(fail to stamp | Tom) = 1/100 = 0.01
P(fail to stamp | Jeff) = 1/60
P(fail to stamp | Pat) = 1/200 = 0.005

Using law of total probability, we have here:
P(fail to stamp) = P(fail to stamp | John)P(John) + P(fail to stamp | Tom) P(Tom) + P(fail to stamp | Jeff)P(Jeff) + P(fail to stamp | Pat)P(Pat)

= 0.005*(0.3 + 0.05) + 0.01*0.5 + (1/60)*0.15 = 0.00475

Therefore, using Bayes theorem now, probability that the non stamped one was from John inspection is computed here as:
P(John | fail to stamp) = P(fail to stamp | John)P(John) / P(fail to stamp)

= 0.005*0.3 / 0.00475

= 0.3158

Therefore 0.3158 is the required probability here.