Fundamentals of Probability and Statistics for Engineers

Theorem 2. 2: Bay es’ t heorem. Let A and B be two arbitrary events with

0 and 0. Then:

Combining this theorem with the total probability theorem we have a useful
consequence:

for any i where events Bj represent a set of mutually exclusive and exhaustive
events.
The simple result given by Equation (2.28) is called Bayes’ theorem after the
English philosopher Thomas Bayes and is useful in the sense that it permits us
to evaluate a posteriori probability in terms of a priori information P(B)
and , as the following examples illustrate.

Example 2.12. Problem: a simple binary communication channel carries
messages by using only two signals, say 0 and 1. We assume that, for a given
binary channel, 40% of the time a 1 is transmitted; the probability that a
transmitted 0 is correctly received is 0.90, and theprobability that a transmitted
1 is correctly received is 0.95. D etermine (a) the probability of a 1 being
received, and (b) given a 1 is received, theprobability that 1 was transmitted.
Answer: let

event that 1 is transmitted

event that 0 is transmitted

event that 1 is received

event that 0 is received

The information given in the problem statement gives us

and these are represented diagrammatically in Figure 2.7.
For part (a) we wish to find P(B). Since A andA are mutually exclusive and
exhaustive, it follows from the theorem of total probability [Equation (2.27)]

Basic Probability Concepts 25

P A)6ˆ P B)6ˆ

P…BjA†ˆ

P…AjB†P…B†

P…A†

:… 2 : 28 †

P…BijA†ˆP…AjBi†P…Bi†

,

Xn

jˆ 1

P…AjBj†P…Bj†



: … 2 : 29 †

P BjA)

P AjB),

Aˆ

Aˆ

Bˆ

Bˆ

P…A†ˆ 0 : 4 ;

P…BjA†ˆ 0 : 95 ;

P…BjA†ˆ 0 : 90 ;

P…A†ˆ 0 : 6 ;

P…BjA†ˆ 0 : 05 ;

P…BjA†ˆ 0 : 10 :

;

;

;

: