The binomial distribution calculates the probability of x successes out of n trials with probability of success p. Mean is np and variance is np(1-p). It belongs to a family of distributions including the Bernoulli and negative binomial. The binomial distribution can be approximated by a normal or Poisson distribution for large n.
A binomial distribution with parameters (n,p) gives the discrete probability of having x successes out of n trials, with the probability of success p, assuming that each trial is independent and that the outcome of a trial is either a success or a failure. The mean number of successes out of n trials is the mean np and the variance is np(1-p). The binomial belongs to a family of event-related distributions that includes the negative binomial and the Bernoulli distribution. Because the probability of the binomial distribution is calculated using the factorial function, which becomes very large as the number of trials increases, the binomial distribution approximation of a normal or Poisson distribution is typically used.
For example, a fair coin is tossed twice and a success is defined as getting heads. The number of trials is n = 2 and the probability of tossing heads is p = ½. The results can be summarized in a binomial distribution table: the probability of no heads, P(x = 0) is 25%, the probability of one heads, P(x = 1) is 50%, and the probability of two heads P(x = 2) is 25%. The expected number of heads tossed is np = 2*1/2 = 1. The variance is np(1-p) = ½.
Other distributions describe the probability of events and belong to the same family as the binomial. A Bernoulli distribution gives the probability of a single event succeeding and is equivalent to a binomial with n = 1. The negative binomial distribution gives the probability of having x failures, where like the regular binomial it gives the probability of x successes.
Often the cumulative density function of the binomial distribution is used, which gives the probability of having x or fewer successes in n trials. Calculating this probability is easy for a small n, but becomes tedious as n gets large, due to the binomial coefficient. The binomial coefficient reads “n choose x” and refers to the number of combinations that x outcomes can be chosen from n possibilities. It is calculated using the factorial function. As the number of trials (n) becomes greater than 70, n factorial becomes huge and can no longer be calculated on a standard calculator.
The approximation of the binomial distribution as n becomes large can be discrete or continuous. If n is very large and p is very small, then the binomial distribution becomes a discrete Poisson distribution. If n is large enough without any constraints on p, then the binomial normal distribution approximation can be used. The binomial mean and standard deviation become the parameters of the normal distribution, and a correction for continuity is applied when calculating the cumulative density function.
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