Statistics for Maintenance Modelling: Basic Probability Concepts

# Rule of Total Probability¤

The use of conditional probability in the last example can be generalised for the situation where the intermediate stage permits k different alternatives (whose occurrence is denoted by B1, B2,…,Bk). It requires the following theorem, sometimes called the rule of total probability:

Have a go at this question. A tool manufacturer rents machines from three other companies: 60% from company 1, 30% from company 2 and 10% from company 3. If 9% of the machines from company 1 need a service, 20% of the machines from company 2 need a service and 6% of the machines from company 3 need a service, what is the probability that one of the machines will need servicing? We'll call this event A.

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| P(A) =

Yes, that's right. If B1, B2, and B3 are the events that the machines come from company 1, 2, or 3 respectively, we can substitute the probabilities into the rule of total probability to give:

P(A) = (0.0)(0.9) + (0.3)(0.2) + (0.1)(0.06) = 0.12

No, P(A) = 0.12. If B1, B2, and B3 are the events that the machines come from company 1, 2, or 3 respectively, we can say that
P(B1) = 0.6, P(B2) = 0.3, P(B3) = 0.1, P(A|B1) = 0.09, P(A|B2) = 0.2, P(A|B3) = 0.06. Using the rule of total probability we get:

P(A) = (0.0)(0.9) + (0.3)(0.2) + (0.1)(0.06) = 0.12