Page 42 - Fundamentals of Probability and Statistics for Engineers

Basic Probability Concepts 25

Theorem 2.2: Bayes’ theorem. Let A and B be two arbitrary events with

P A) 6ˆ 0 and P B) 6ˆ 0. Then:
P AjB†P B†
P BjA†ˆ : 2:28†
P A†
Combining this theorem with the total probability theorem we have a useful
consequence:
n
,
P B i jA†ˆ P AjB i †P B i † X P AjB j †P B j † : 2:29†

jˆ1
for any i where events B j 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 P BjA) in terms of a priori information P(B)
and P AjB), , 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 the probability that a transmitted
1 is correctly received is 0.95. Determine (a) the probability of a 1 being
received, and (b) given a 1 is received, the probability that 1 was transmitted.
Answer: let

A ˆ event that 1 is transmitted;
A ˆ event that 0 is transmitted;
B ˆ event that 1 is received;
B ˆ event that 0 is received:

The information given in the problem statement gives us

P A†ˆ 0:4; P A†ˆ 0:6;
P BjA†ˆ 0:95; P BjA†ˆ 0:05;
P BjA†ˆ 0:90; P BjA†ˆ 0:10:

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

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