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: $$A, \overline{A}$$ (success and defeat).
  • the probabilities of every event $$A, \overline{A}$$ 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.

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:

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

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:

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

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

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

number of experiments: $$n=10$$

number of successful results: $$k=8$$

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

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

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:

$$$\mu= n \cdot p$$$

The variance is:

$$$\sigma^2= n \cdot p \cdot q= n \cdot p \cdot (1-p)$$$

The standard deviation is:

$$$\sigma = \sqrt{n\cdot p \cdot q}$$$