A binomial random variable is a random variable that counts the results of n
"binomial trials." A binomial trial is simply a much simpler random variable that takes the value
1 with probability p and takes the value 0 with probability 1−p.
The simplest type of binomial trial is the flip of a fair coin (which corresponds to p=1/2).
For this reason, other binomial trials may be thought of as weighted coin flips.
For n binomial trials with probability p, a binomial random variable Bin(n,p)
counts the number of trials taking the value 1. One may verify that if X is such
a random variable, then Pr(X=k) is given by \binom{n}{k} p^k \cdot (1-p)^{n-k}.
The chart below illustrates three different binomial random variables over 20 trials.
In each case, k is plotted against \mathrm{Pr}(X = k) as k ranges between
0 and 20. The blue chart represents p = 0.1, the green chart represents p = 0.5, and the
red chart represents p = 0.8.