Probability of a Defective Product — the Law of Total Probability

Problem. A certain product is made by two factories a and b. Suppose 18% of the products of factory a and 12% of the products of factory b are defective. The products of the two factories are mixed in the ratio a : b = 4 : 6 into a very large batch, and one product is drawn at random from it. What is the probability that it is defective?

Move the parameters

Let A be the event that the product is from factory a, B the event that it is from factory b, and C the event that it is defective; compute P(C) using A and B.

Defect rate of factory a P(C|A) = 18%

Defect rate of factory b P(C|B) = 12%

Mixing ratio a : b = 40 : 60

P(A) = 0.40, P(B) = 0.60

Area diagram — probabilities as areas

factory a, good factory a, defective P(A∩C) factory b, good factory b, defective P(B∩C)

Column widths are the mixing proportions P(A), P(B); the heights of the dark bands are the defect rates. The total area of the two dark rectangles is P(C).

Computation — the law of total probability

0.144 = probability that the drawn product is defective, P(C)

Extension (Bayes' theorem): given that the product is defective, which factory did it come from?

Simulation — check it by actually drawing products

0 products drawn

	good	defective	total
factory a	0	0	0
factory b	0	0	0
total	0	0	0

Observed fraction of defectives

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Theoretical value P(C)

0.144

Origin of the defective draws (observed) → P(A|C), P(B|C)

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Theoretical values (Bayes' theorem)