Fundamentals of Probability and Statistics for Engineers

D ef inition 2. 3. The conditional probability of A given that B has occurred is
given by

Definition 2.3 is meaningless if P(B) 0.
It is noted that, in the discussion of conditional probabilities, we are dealing
with a contracted sample space in which B is known to have occurred. In other
words, B replaces S as the sample space, and the conditional probability P(AB)
is found as the probability of A with respect to this new sample space.
In the event that A and B are independent, it implies that the occurrence of B
has no effect on the occurrence or nonoccurrence of A. We thus expect
and Equation (2.24) gives

or

which is precisely the definition of independence.
It is also important to point out that conditional probabilities areprobabilities
(i.e. they satisfy the three axioms of probability). U sing Equation (2.24), we see that
the first axiom is automatically satisfied. For the second axiom we need to show that

This is certainly true, since

As for the third axiom, if A 1 ,A 2 ,... are mutually exclusive, then A 1 B, A 2 B,...
are also mutually exclusive. H ence,

Basic Probability Concepts 21

P…AjB†ˆ

P…AB†

P…B†

; P…B†6ˆ 0 : … 2 : 24 †

ˆ

j

P AjB)ˆP A),

P…A†ˆ

P…AB†

P…B†

;

P…AB†ˆP…A†P…B†;

P…SjB†ˆ 1 :

P…SjB†ˆ

P…SB†

P…B†

ˆ

P…B†

P…B†

ˆ 1 :

P‰…A 1 [A 2 [...†jBŠˆ

P‰…A 1 [A 2 [...†BŠ

P…B†

ˆ

P…A 1 B[A 2 B[...†

P…B†

ˆ

P…A 1 B†

P…B†

‡

P…A 2 B†

P…B†

‡

ˆP…A 1 jB†‡P…A 2 jB†‡;