Introduction to Sample Space and Probability - Random Signals and Processes Primer with MATLAB - page 7

Fig. 1.5 Illustration of the subset of A

P.5 If events A and B are not mutually exclusive, Fig. 1.6 (i.e., their intersection is
not an empty set) then:

PfA [ Bg¼PfAgþPfBgPfA \ Bg:

(1.24)

Proof To derive ( 1.24 ) we must first note that A[B can be written as the union of
three disjoint events (Fig. 1.6 ):

A \ B ð The points in A which are not in BÞ;

A \ B ð The points in B which are not in AÞ;

A \ B ð The points in both ; A and BÞ:

A [ B ¼ðA \ BÞ[ðA \ BÞ[ðA \ BÞ:

(1.25)

Thus, from Axiom III, and ( 1.25 ) we have:

PfA [ Bg¼PfA \ BgþPfA \ BgþPfA \ Bg:

(1.26)

Fig. 1.6 Events A and B are not mutually exclusive