Content

 Introduction

 What is probability?

 Combining probabilities

 The two-state system

 Combinatorial analysis

 The binomial distribution

It follows from Eqs. (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
\begin{displaymath} P_N(n_1) = \frac{N!}{n_1 !\,(N-n_1)!} \,p^{n_1}q^{N-n_1}. \end{displaymath} (21)

This probability function is called the binomial distribution function. The reason for this is obvious if we tabulate the probabilities for the first few possible values of $N$.
 
        $n_1$    
    0 1 2 3 4
  1 $q$ $p$      
$N$ 2 $q^2$ $2pq$ $p^2$    
  3 $q^3$ $3pq^2$ $3p^2q$ $p^3$  
  4 $q^4$ $4pq^3$ $6p^2q^2$ $4p^3q$ $p^4$

 
 
Of course, we immediately recognize these expressions: 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 can be written
\begin{displaymath} (p+q)^N \equiv \sum_{n=0}^{N} \frac{N!}{n!\,(N-n)!}\,p^n q^{N-n}. \end{displaymath} (22)

Equations (2.21) and (2.22) can be used to establish the normalization condition for the binomial distribution function:
\begin{displaymath} \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, \end{displaymath} (23)

since $p+q=1$.

 The mean, variance, and standard deviation

 Application to the binomial distribution

 The Gaussian distribution

 The central limit theorem