Four workers at a fast food restaurant pack the takeout chic

Four workers at a fast food restaurant pack the take-out chicken dinners. John packs 45% of the dinners but fails to include a salt packet 4% of the time. Mary packs 25% of the dinners but omits the salt 2% of the time. Sue packs 30% of the dinners but fails to include the salt 3% of the time. You have purchased a dinner and there is no salt.

a. Find the probability that John packed your dinner.

b. Find the probability that Mary packed your dinner.

Solution

This scenario is a perfect application of Bayes\' Theroem. We have:
P(John) = .45
P(Mary) = .25
P(Sue) = .30
P(NoSalt|John) = .04
P(NoSalt|Mary) = .02
P(NoSalt|Sue) = .03

In part a, we want to know P(John|NoSalt). Applying Bayes\':
P(John|NoSalt) = P(NoSalt|John)P(John) / P(NoSalt|John)P(John)+P(NoSalt|Mary)P(Mary)+P(NoSalt|Sue)P(Sue)
= (.04)(.45) / (.04)(.45) + (.02)(.25) + (.03)(.30)
= 0.018 / (0.018 + 0.005 + 0.009)
= 0.5625

In part b, we want to know P(Mary|NoSalt). Again, applying Bayes\' in the same way:
P(Mary|NoSalt) = P(NoSalt|Mary)P(Mary) / P(NoSalt|John)P(John)+P(NoSalt|Mary)P(Mary)+P(NoSalt|Sue)P(Sue)
= (.02)(.25) / (.04)(45) + (.02)(.25) + (.03)(.30)
= 0.5 / (1.8 + 0.5 + 0.9)
= 0.15625