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.

Step-by-step explanation:

This scenario is a perfect application of Bayes' Theorem. We have:

P (John) =.45

P (Mary) =.25

P (Sue) =.30

P (No Salt|John) =.04

P (No Salt|Mary) =.02

P (No Salt|Sue) =.03

In part a, we want to know P (John|No Salt). Applying Bayes:

P (John|No Salt) = P (No Salt|John) P (John) / P (No Salt|John) P (John) + P (No Salt|Mary) P (Mary) + P (No Salt|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|No Salt). Again, applying Bayes' in the same way:

P (Mary|No Salt) = P (No Salt|Mary) P (Mary) / P (No Salt|John) P (John) + P (No Salt|Mary) P (Mary) + P (No Salt|Sue) P (Sue)

= (.02) (.25) / (.04) (45) + (.02) (.25) + (.03) (.30)

= 0.5 / (1.8 + 0.5 + 0.9)

= 0.15625