The binomial distribution gives the probability of k successes in n independent trials that have a yes or no answer, known as Bernoulli trials, where p is the probability of success. The binomial distribution can be used in genetics to determine the probability the k out of n individuals will have a particular genotype. In this case, having that particular genotype is considered "success."
The binomial distribution is given by,
where P (k/n) is the probability of k successes in n trials, and p is the probability of a success. Recall that m! = m · (m - 1) · (m - 2) · · · 2 · 1 where m is a positive integer, and 0! = 1. Because n trials are yes/no, notices that there are k successful trials, each with probability p, and the remaining n - k trials are failures, each with probability 1 - p.
To demonstrate the binomial distribution, let n = 5 and k = 2, in other words there are 2 successes in 5 trials. Under these circumstances the distribution becomes,
where p is the probability of success.