Cascade Paint mixes paint in three separate plants and then

Cascade Paint mixes paint in three separate plants and then ships the unmarked cans to central warehouse. Plant A supplies 50% of the paint, and past records indicate that the paint is incorrectly mixed 10% of the time. Plant B contributes 30% with a defective rate of 5%. Plant C supplies 20% with paint mixed incorrectly 20% of the time. Cascade guarantees its product and spent $10,000 replacing improperly mixed paint last year. How should the cost be distributed among the three plants? Using Bayes\' Theorem.

Solution

For Plant A:

Probability of having defective product is P(Good Product)x P(Wrong Product)

= P(0.5) x P(0.1) = 0.05

For Plant B:

Probability of having defective product is P(0.3) x P(0.05) = 0.015

For Plant C:

Probability of having defective product is P(0.2) x P(0.2) = 0.04

Plant A has a percentage of defective product from total supply is:

0.05 / (0.05+0.015+0.04) = 0.05/0.105 = 0.4761

Plant B has a percentage of defective product from total supply is:

0.015 / (0.05+0.015+0.04) = 0.05/0.105 = 0.1428

Plant C has a percentage of defective product from total supply is:

0.04 / (0.05+0.015+0.04) = 0.05/0.105 = 0.3809

Plant A has the distributed cost of 0.4761 x $10,000 = $4761

Plant B has the distributed cost of 0.1428 x $10,000 = $1428

Plant C has the distributed cost of 0.3809 x $10,000 = $3809