Page 32 - Fundamentals of Probability and Statistics for Engineers

Basic Probability Concepts 15

B

A

Figure 2.5 Venn diagram for derivation of Equation (2.12)

Hence, using Axiom 3,

P A [ B†ˆ P A ‡ AB†ˆ P A†‡ P AB†: 2:13†

Furthermore, we note

AB ‡ AB ˆ B:

Hence, again using Axiom 3,

P AB†‡ P AB†ˆ P B†;

or

P AB†ˆ P B† P AB†:

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 computing P(A [ B), the probability associated with
one AB must be subtracted from P(A) ‡ P(B), giving Equation (2.12) (see
Figure 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,

P A [ B [ C†ˆ P A†‡ P B†‡ P C† P AB† P AC†
2:14†
P BC†‡ P ABC†:

TLFeBOOK