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PQ220 6234F.Ch 03 13/04/2002 03:19 PM Page 73

3-6 BINOMIAL DISTRIBUTION 73

and by definition 0! 1 . We also use the combinatorial notation

n n!
a b
x x! 1n x2!
For example,

5 5! 120
a b 10
2 2! 3! 2 6
See Section 2-1.4, CD material for Chapter 2, for further comments.

EXAMPLE 3-16 The chance that a bit transmitted through a digital transmission channel is received in error is
0.1. Also, assume that the transmission trials are independent. Let X the number of bits in
error in the next four bits transmitted. Determine P1X 22 .
Let the letter E denote a bit in error, and let the letter O denote that the bit is okay, that is,
received without error. We can represent the outcomes of this experiment as a list of four let-
ters that indicate the bits that are in error and those that are okay. For example, the outcome
OEOE indicates that the second and fourth bits are in error and the other two bits are okay. The
corresponding values for x are

Outcome x Outcome x

OOOO 0 EOOO 1
OOOE 1 EOOE 2
OOEO 1 EOEO 2
OOEE 2 EOEE 3
OEOO 1 EEOO 2
OEOE 2 EEOE 3
OEEO 2 EEEO 3
OEEE 3 EEEE 4

The event that X 2 consists of the six outcomes:

5EEOO, EOEO, EOOE, OEEO, OEOE, OOEE6

Using the assumption that the trials are independent, the probability of {EEOO} is

2
2
P1EEOO2 P1E2P1E2P1O2P1O2 10.12 10.92 0.0081
Also, any one of the six mutually exclusive outcomes for which X 2 has the same proba-
bility of occurring. Therefore,

P1X 22 610.00812 0.0486

In general,

x
P1X x2 (number of outcomes that result in x errors) times 10.12 10.92 4 x