# The Binomial Distribution

The binomial distribution is the last of the three "basic distributions". I say basic here because together with the Poisson and the Gaussian distribution the binomial distribution appears very often in very many places. The binomial distribution describes how probable it is to get exactly \(k\) events of probability \(p\) in \(n\) tries if there are only two outcomes (either something happens or it does not happen). The Binomial distribution is written as:

$$ P(k,n,p) = \binom{n}{k} \cdot p^k(1-p)^{n-k} $$

Where \( \binom{n}{k} \) is the binomial coefficient which is defined as

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

It should be noted that \( k \) can't get larger than \( n \) as one observation can never generate more than one result (one coin flip will never generate more than one heads or tails). As \( n \) is the number of tries and \( k \) the number of results, both \( n \) and \( k \) must be positive integers to make sense.