Binomial Probability Distribution

It follows from Equations (2.16) and (2.20) that the probability of obtaining $n_1$ occurrences of the outcome 1 in $N$ statistically independent observations of a two-state system is

$$P_N(n_1) = \frac{N!}{n_1 ! (N-n_1)!} p^{ n_1} q^{ N-n_1}.$$ (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 $N$ , 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 $(p+q)$ , $(p+q)^{ 2}$ , $(p+q)^{ 3}$ , and $(p+q)^{ 4}$ , respectively. In algebra, the expansion of $(p+q)^{ N}$ is called the binomial expansion (hence, the name given to the probability distribution function), and is written

$$(p+q)^{ N} \equiv \sum_{n=0,N}\frac{N!}{n! (N-n)!} p^{ n} q^{ N-n}.$$ (2.22)

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

$$\sum_{n_1=0,N} P_N(n_1) =\sum_{n_1=0,N} \frac{N!}{n_1! (N-n_1)!} p^{ n_1} q^{ N-n_1}\equiv (p+q)^{ N} = 1,$$ (2.23)

because $p+q=1$ . [See Equation (2.11).]

Table 2.1: The binomial probability distribution, $P_N(n_1)$ .

				$n_1$
		0	1	2	3	4
	1	$q$	$p$
$N$	2	$q^{ 2}$	$2 p q$	$p^{ 2}$
	3	$q^{ 3}$	$3 p q^{ 2}$	$3 p^{ 2} q$	$p^{ 3}$
	4	$q^{ 4}$	$4 p q^{ 3}$	$6 p^{ 2} q^{ 2}$	$4 p^{ 3} q$	$p^{ 4}$