Page 44 - Fundamentals of Probability and Statistics for Engineers

Basic Probability Concepts 27

0.4 0.95 0.95
A 0.05 B 0.05 C

0.1 0.1

0.6 0.9 0.9
A B C

Figure 2.8 A two-stage binary channel

by adding a second stage to the communication channel, with Figure 2.8
showing all the associated probabilities. We wish to determine P(C), the prob-
ability of receiving a 1 at the second stage.
Tree diagrams are useful for determining the behavior of this system when the
system has a ‘one-stage’ memory; that is, when the outcome at the second stage is
dependent only on what has happened at the first stage and not on outcomes at
stages prior to the first. Mathematically, it follows from this property that

P CjBA†ˆ P CjB†; P CjBA†ˆ P CjB†; etc: 2:30†

The properties described above are commonly referred to as Markovian
properties. Markov processes represent an important class of probabilistic
process that are studied at a more advanced level.
Suppose that Equations (2.30) hold for the system described in Figure 2.8.
The tree diagram gives the flow of conditional probabilities originating from
the source. Starting from the transmitter, the tree diagram for this problem has
the appearance shown in Figure 2.9. The top branch, for example, leads to the
probability of the occurrence of event ABC , which is, according to Equations
(2.26) and (2.30),

P ABC†ˆ P A†P BjA†P CjBA†
ˆ P A†P BjA†P CjB†
ˆ 0:4 0:95† 0:95†ˆ 0:361:

The probability of C is then found by summing the probabilities of all events
that end with C. Thus,

P C†ˆ P ABC†‡ P ABC†‡ P ABC†‡ P A BC†
ˆ 0:95 0:95† 0:4†‡ 0:1 0:05† 0:4†‡ 0:95 0:1† 0:6†‡ 0:1 0:9† 0:6†
ˆ 0:472:

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