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. |