Page 38 - Fundamentals of Probability and Statistics for Engineers

Basic Probability Concepts 21

Definition 2.3. The conditional probability of A given that B has occurred is

given by
P AB†
P AjB†ˆ ; P B† 6ˆ 0: 2:24†
P B†
Definition 2.3 is meaningless if P(B) ˆ 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
j
words, B replaces S as the sample space, and the conditional probability P(A B)
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
P AjB) ˆ P A), and Equation (2.24) gives

P AB†
P A†ˆ ;
P B†
or
P AB†ˆ P A†P B†;

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

P SjB†ˆ 1:

This is certainly true, since
P SB† P B†
P SjB†ˆ ˆ ˆ 1:
P B† P B†
As for the third axiom, if A 1 , A 2 , .. . are mutually exclusive, then A 1 B, A 2 B,...
are also mutually exclusive. Hence,
P‰ A 1 [ A 2 [ ...†BŠ
P‰ A 1 [ A 2 [ ...†jBŠˆ
P B†
P A 1 B [ A 2 B [ ...†
ˆ
P B†
P A 1 B† P A 2 B†
ˆ ‡ ‡
P B† P B†
ˆ P A 1 jB†‡ P A 2 jB† ‡ ;

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