In the theory of probability and statistics, a Bernoulli trial (or binomial trial) is a random experiment with exactly two possible outcomes, "success" and "failure", in which the probability of success is the same every time the experiment is conducted. Wikipedia
When we want to know the probability of k successes in n such trials, we should look into binomial distribution.
When we want to know the probability of getting the first success on k-th trial, we should look into geometric distribution.
When we want to know the probability that the k-th success is observed on the n-th trial, we should look into negative binomial distribution.
Probability density function of negative binomial distribution is
- p is the probability of success of a single trial,
- x is the trial number on which the k-th success occurs.
- is the number of combinations of m from n
Cumulative distribution function of negative binomial distribution is
- is the regularized incomplete beta function
Note that , that is, the chance to get the k-th success on the k-th trial is exactly k multiplications of p, which is quite obvious.
Mean or expected value for the negative binomial distribution is
The calculator below calculates the mean and variance of the negative binomial distribution and plots the probability density function and cumulative distribution function for given parameters: the probability of success p, number of successes k, and the number of trials to plot on chart n.
Note that there are other formulations of the negative binomial distribution. They are created using the following notation: n - number of trials, r - number of failures, k - number of successes, with n=k+r. These are:
- k successes, given r failures
- n trials, given r failures
- r failures, given k successes
- n trials, given k successes (a case described above)
- k successes, given n trials (binomial distribution).
They have slightly different formulas.