Binomial distribution

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In probability theory and statistics, the binomial distribution is the discrete probability distribution of the number of successes in a sequence of n independent yes/no experiments, each of which yields success with probability p. Such a success/failure experiment is also called a Bernoulli experiment or Bernoulli trial. In fact, when n = 1, the binomial distribution is a Bernoulli distribution. The binomial distribution is the basis for the popular binomial test of statistical significance.

It is frequently used to model number of successes in a sample of size n from a population of size N. If the samples are not independent (this is sampling without replacement), the resulting distribution is a hypergeometric distribution, not a binomial one. However, for N much larger than n, the binomial distribution is a good approximation, and widely used.

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Examples

An elementary example is this: roll a standard die ten times and count the number of fours. The distribution of this random number is a binomial distribution with n = 10 and p = 1/6.

As another example, flip a coin three times and count the number of heads. The distribution of this random number is a binomial distribution with n = 3 and p = 1/2.