B. P. Lathi, Zhi Ding - Modern Digital and Analog Communication Systems-Oxford University Press (2009)

404 FUNDAMENTALS OF PROBABILITY THEORY
For instance. if P e = 10 - 4. P(E) :::::: 3 x 10 -8 . Thus, the error probability is reduced
from 1 o -4 to 3 x 1 o -8 . We can use any odd number of repetitions for this scheme to
function.
In this example, higher reliability is achieved at the cost of a reduction in the rate of
information transmission by a factor of 3. We shall see in Chapter 14 that more efficient
ways exist to effect a trade-off between reliability and the rate of transmission through the
use of error correction codes.
Multiplication Rule for Conditional Probabilities
As shown in Eq. (8.12), we can write the joint event
P(A n B) = P(A)P(B/A)
This rule on joint events can be generalized for multiple events A 1, A2, ... , A n via iterations.
IfAIA2 • •-A ,, =f. 0, then we have
P(AIA2 · ··A n ) P(AIA2 · ··A n-I ) P(AIA2)
P(AIA2 · ··A ,, ) = ----- · ----- · · · --- · P(AI )
P(AIA2 · ··A n-I ) P(A IA2 · · · An-2) P(A1 )
(8.1 8a)
= P(A n lA1A2 · · · A n-1 ) · P (A n- 1 IAIA2 · · · An-2) · .. P(A2IA1 ) · P(A 1 )
(8.18b)
Note that sinceAIA2 • ••A n =f. 0, every denominator in Eq. (8.1 8a) is positive and well defined.
Example 8. 9
Suppose a box of diodes consist of N g
good diodes and Nb bad diodes. If five diodes are
randomly selected, one at a time, without replacement, determine the probability of obtaining
the sequence of diodes in the order of good, bad, good, good, bad.
We can denote Gi as the event that the ith draw is a good diode. We are interested in the
event of G1 GG3 G 4 G5.
N g
N g
+ Nb
Nb - I
Nb
N g
+ Nb - I
N g
- I
.
N g
- 2
Nb + N g - 2 N g
+ Nb - 3
To Divide and Conquer: The Total Probability Theorem
In analyzing a particular event of interest, sometimes a direct approach to evaluating its probability
can be difficult because there can be so many different outcomes to enumerate. When
dealing with such problems, it is often advantageous to adopt the divide-and-conquer approach
by separating all the possible causes leading to the particular event of interest B. The total
probability theorem provides a perfect tool for analyzing the probability of such problems.
We define S as the sample space of the experiment of interest. As shown in Fig. 8.4, the
entire sample space can be partitioned into n disjoint events A1 , ... , A n . We can now state the
theorem: