Binomial Distribution

The binomial distribution is the number of times some outcome occurs out of n independent trials, each having the same probability of “success” p.

The PMF is :

P (X = k) = (\binom{n}{k}) p^{k} (1 - p)^{n - k}, k = 0, 1, \dots, n .

If n is the number of objects in a set and k is the subset. The binomial coefficient is

(\binom{n}{k}) = \frac{n!}{(n - k)! k!},

where we read this as " $n$ choose $k$ "

Assumptions

Each trial is independent, the probability of success is the same for every trial, and the number of trials is fixed in advance.

Properties

Mean: $n p$
Variance: $n p (1 - p)$

You can have a normal approximation Normal Distribution to the binomial theorem (under specific conditions) through the
The distribution converges to a Poisson Distributions as n increases, p goes to infinity, and np approaches $λ$