The binomial (or Bernoulli) distribution

An experiment can be modeled with a binomial distribution whenever:

there are only two possible events resulting from the experiment: $A, \overset{―}{A}$ (success and defeat).
the probabilities of every event $A, \overset{―}{A}$ are the same in any happening of the experiment ( $p$ and $q = 1 - p$ , respectively). Namely if a coin is flipped several times, the probability of having 'heads' does not change.
any realization of the experiment is independent from the rest.

A binomial random variable will give the number of successes when having happened a certain number of experiments.

Example

It turns out to be useful to analyze the number of times that 'heads' is obtained when flipping a coin $n$ times.

The binomial distribution is usually represented by $B (n, p)$ , with:

$n$ : number of happenings of the random experiment.
$p$ : probability of success in doing an experiment

So if we want to study the binomial distribution that models $10$ flips of a coin (in which the 'heads' and 'tails' are equally probable) we have:

$B (10, \frac{1}{2})$

The probability function of the binomial distribution is:

$p (X = k) = (\binom{n}{k}) p^{k} \cdot q^{n - k}$

$n$ : number of experiments
$k$ : number of successes
$p$ : success probability
$q$ : defeat probability

The combinatorial number is defined:

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

Example

Calculate the probability of obtaining $8$ 'heads' when flipping a coin ten times.

Distribution $B (10, \frac{1}{2})$

number of experiments: $n = 10$

number of successful results: $k = 8$

probability of each success and each defeat: $p = q = 1 / 2$

$p (X = 8) = (\binom{10}{8}) (\frac{1}{2})^{8} (\frac{1}{2})^{2} = 0.044$

what can be interpreted as the product of the possible combinations of $8$ 'heads' and $2$ 'tails' times the probability of extracting $8$ 'heads' times the probability of extracting $2$ 'tails'.

The average of a binomial distribution is:

$μ = n \cdot p$

The variance is:

$σ^{2} = n \cdot p \cdot q = n \cdot p \cdot (1 - p)$

The standard deviation is:

$σ = \sqrt{n \cdot p \cdot q}$