Faulty machine? A Bayesian posterior from one defect

Question

A machine on a line is either healthy or faulty. From experience, 20 percent of machines are faulty. A faulty machine produces a defective item with probability 0.4, while a healthy machine produces a defective item with probability 0.05. You inspect one item from a machine and it is defective.

Using Bayes' rule, what is the probability that the machine is faulty?

Accepted Answer

Intuition first. A faulty machine is eight times more likely to make a defect than a healthy one, so a defect points toward "faulty." But only 20 percent of machines are faulty to begin with, so we must fold that base rate in. Bayes' rule does the weighing. The recipe. Multiply each hypothesis' prior by the chance it produces the defect, then normalize. The two weighted paths: 	ext{faulty path} = P(	ext{faulty})\cdot P(	ext{defect}\mid	ext{faulty}) = 0.20	imes 0.40 = 0.08, 	ext{healthy path} = P(	ext{healthy})\cdot P(	ext{defect}\mid	ext{healthy}) = 0.80	imes 0.05 = 0.04. Normalize …

Faulty machine? A Bayesian posterior from one defect

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