Fundamentals of Probability and Statistics for Engineers

Hence, using Axiom 3,

Furthermore, we note

Hence, again using Axiom 3,

or

Substitution of this equation into Equation (2.13) yields Equation (2.12).
Equation (2.12) can also be verified by inspecting the Venn diagram in Figure
2.5. The sum P(A) P(B) counts twice the events belonging to the shaded
area AB. Hence, in co mputing P(A B), the probability associated with
one AB must be subtracted from P(A) P(B), giving Equation (2.12) (see
F igure 2.5).
The important result given by Equation (2.12) can be immediately general-
ized to the union of three or more events. Using the same procedure, we can
show that, for arbitrary events A, B, and C,

A

B

Figure 2.5 Venn diagram for derivation of Equation (2.12)

Basic Probability Concepts 15

P…A[B†ˆP…A‡AB†ˆP…A†‡P…AB†: … 2 : 13 †

AB‡ABˆB:

P…AB†‡P…AB†ˆP…B†;

P…AB†ˆP…B†P…AB†:

‡

[

‡

P…A[B[C†ˆP…A†‡P…B†‡P…C†P…AB†P…AC†

P…BC†‡P…ABC†:

… 2 : 14 †