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Binomial Probability Distribution
It follows from Equations (2.16) and (2.20) that the probability of obtaining
occurrences of the
outcome 1 in
statistically independent observations of a twostate system
is

(2.21) 
This probability function is called the binomial probability distribution.
The reason for this name becomes obvious if we tabulate the probabilities for the
first few possible values of
, as is done in Table 2.1.
Of course, we immediately recognize the expressions appearing in the first four rows of this table:
they appear in the standard
algebraic expansions of
,
,
, and
,
respectively. In algebra, the expansion of
is called the
binomial expansion (hence, the name given to the probability distribution
function),
and is written

(2.22) 
Equations (2.21) and (2.22) can be used to establish the normalization
condition for the binomial distribution function:

(2.23) 
because
. [See Equation (2.11).]
Table 2.1:
The binomial probability distribution,
.

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Richard Fitzpatrick
20160125