Statistics for Maintenance Modelling: Basic Probability Concepts

# Bayes' Theorem¤

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Referring to the previous question, suppose that we are interested to know that if a machine needs a service, what is the probability that it came from company 2? To answer this type of question we need Bayes' Theorem.

If B1, B2,…, Bk constitute a partition of the sample space S and P(Bi) 0 for i = 1, 2, ..., k then for any event A in S such that P(A) 0:

So in the previous example, if a machine needed a service (P(A)), what is the probability that it came from company 2? To recap: P(B1) = 0.6, P(B2) = 0.3, P(B3) = 0.1, P(A|B1) = 0.09,
P(A|B2) = 0.2 and P(A|B3) = 0.06.

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| P(B2|A) =

Yes, that's right. Substituting the probabilities into Bayes' Theorem gives:
P(B2|A) = (0.3)(0.2) / ( (0.6)(0.9) + (0.3)(0.2) + (0.1)(0.06) ) = 0.06 / 0.12 = 0.5
No, P(B2|A) = 0.5. Substituting the probabilities into Bayes' Theorem gives:
P(B2|A) = (0.3)(0.2) / ( (0.6)(0.9) + (0.3)(0.2) + (0.1)(0.06) ) = 0.06 / 0.12 = 0.5