Fundamentals of Probability and Statistics for Engineers

Axiom 1: P(A) 0 (nonnegative).

Axiom 2: P(S) 1 (normed).

Axiom 3: for a countable co llection of mutually exclusive events A 1 ,A 2 ,...inS,

These three axioms define a countably additive and nonnegative set function
P(A), A S. As we shall see, they constitute a sufficient set of postulates from
which all useful properties of the probability function can be derived. Let us
give below some of these important properties.
First, P( ) 0. Since S and are disjoint, we see from Axiom 3 that

It then follows from Axiom 2 that

or

Second, if A C, then P(A) P(C). Since A C, one can write

where B is a subset of C and disjoint with A. Axiom 3 then gives

Since P(B) 0 as required by Axiom 1, we have the desired result.
Third, given two arbitrary events A and B, we have

In order to show this, let us write A B in terms of the union of two
mutually exclusive events. F rom the second relation in Equations (2.10),
we write

14 Fundamentals of Probability and Statistics for Engineers

. 
. ˆ
.

P…A 1 [A 2 [...†ˆP

X

j

Aj

!

ˆ

X

j

P…Aj†…additive†:… 2 : 11 †

;ˆ ;



P…S†ˆP…S‡;†ˆP…S†‡P…;†:

1 ˆ 1 ‡P…;†

P…;†ˆ 0 :

  

A‡BˆC;

P…C†ˆP…A‡B†ˆP…A†‡P…B†:



P…A[B†ˆP…A†‡P…B†P…AB†: … 2 : 12 †

[

A[BˆA‡AB: