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 $\mathrm{Bin}(n, p)$
counts the number of trials taking the value 1. One may verify that if $\mathrm{X}$ is such
a random variable, then $\mathrm{Pr}(X = k)$ 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 $p = 0.8$.