Statistics for Maintenance Modelling: Basic Probability Concepts

# Introduction to Bayes' Theorem¤

The definition of conditional probability can be taken a step further to give Bayes' Theorem. Bayes' Theorem is useful when calculating the probability of a certain outcome for an experiment when the outcome depends on various intermediate stages. To understand Bayes' Theorem it is useful to look at a few examples.

Suppose the completion of a building renovation may be delayed because of a strike. The probabilities are:

- 0.4 that there will be a strike
- 0.9 that the renovation will be completed on time if there is no strike
- 0.3 that the renovation will be completed on time if there is a strike.

What is the probability that the renovation job will be completed on time?

If A is the event that the renovation will be completed on time and B is the event that there will be a strike, we are given P(B) = 0.4, P(A|B) = 0.9 and P(A|B) = 0.3. We can write:

P(A) = P((AB)(AB)) (click here for an explanation)

= P(AB) + (AB) (from the definition of a union)

= P(B) P(A|B) + P(B) P(A|B) (from the definition of conditional probability)

Substituting, gives: P(A) = (0.4)(0.3) + (1 - 0.4) (0.9) = 0.66

If P(F) > 0, the definition of conditional probability is:
. Therefore, P(EF) = P(F) P(E|F).