5.2 Binomial distribution for proportions (discrete)
Deriving the binomial distribution is best done using an example.
Example 5.1 (Deriving the binomial distribution) Suppose we know that the probability of finding a cracked egg is \(p=0.1\). If we have a set of, say, \(n = 12\) independent eggs, what is the probability that three will be cracked? Well, three are cracked with probability \(0.1^3\), and nine are not cracked with probability \((1- 0.1)^{12-3}\).
In addition, there are lots of ways to pick which three of the twelve eggs may be the ones that are cracked. We could pick any of the twelve eggs to be the first one; then any of the remaining 11 for the second; and any of the remaining 10 for the thirds.
And some of these selections are the same (picking Egg 1 first, then Eggs 4 and 5, is the same as selecting Egg 4 first, followed by Eggs 1 and 5); in fact, there \(3! = 6\) ways of picking three eggs to be the cracked ones.
So there are \((12\times 11\times 10)/3!\) ways to pick which three eggs are cracked.
So the probability is \[ \Pr(\text{$k=3$ cracked eggs out of $n=12$}) = {12\choose 3} 0.1^3 (1 - 0.1)^{12-3}. \] Note that \({12\choose 3}\) is ‘12 choose 3,’ and is defined such that \[ {n\choose k} = \frac{n!}{k! (n-k)!}. \]
This leads to the binomial distribution, useful for modelling counts from a set, or proportions.
Definition 5.2 (Binomial distribution) The PMF for a binomial distribution is \[ p_X(x) = {n\choose k} p^x (1-p)^{n-k}\quad\text{for $x=0, 1, 2, \dots, n$}, \] for \(0<p<1\) and \(0\le k \le n\).