A. The Binomial Distribution

This you have seen in CS1021. It is also called a Bernoulli distribution. It is applicable to the situation where there are $N$ independent trials, each of which can have one of two outcomes. Call one of the outcomes $0$ and the other $1$. Let $p$ be the probability that any trial yields the outcome $1$. Then the probability of getting $n$ 1's in $N$ trials is given by

$$P(n\vert p,N) = {N \choose n} p^n (1-p)^{N-n} ,$$

where

$${N \choose n} \equiv \frac{N!}{n!(N-n)!} .$$

The expected mean number of 1's is $N p$; the variance is $N p (1-p)$

In the limit of large $N$ with $p$ fixed, a binomial distribution is well approximated by a continuous Gaussian distribution, with the appropriate mean and variance.