Statistics for Maintenance Modelling: Basic Probability Concepts

# Proof of Proposition 3¤

P(EF) = P(E) + P(F) - P(EF) can be proved using a Venn diagram where (EF) is divided into three mutually exclusive sections as shown:

| Section I represents all the points in E that are not in F (i.e. EF)
Section II represents all the points both in E and in F (i.e. EF)
Section III represents all the points in F that are not in E (i.e EF)

From the diagram, we see that:

EF = I II III

E = I II

F = II III

As I, II, and III are mutually exclusive, it follows from axiom 3 that:

For any sequence of mutually exclusive events E1, E2 etc., the probability of at least one of these mutually exclusive events occurring is just the sum of their respective probabilities.

P(EF) = P(I) + P(II) + P(III)

P(E) = P(I) + P(II)

P(F) = P(II) + P(III)

substituting P(I) and P(III) we see that:

P(EF) = P(E) - P(II) + P(F) - P(II) + P(II) = P(E) + P(F) - P(II)

and since II = (EF), proposition 3 is proved.

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