# Negative binomial distribution Probabilities

Calculates probabilities under a Negative binomial distribution.

The random variable counts the number of failures before n successes are achieved. p is the probability of success of a single trial.

## Negative binomial distribution formulas

X is a random variable that follows a negative binomial distribution. It counts the number of failures before n successes.

p is the success probability of a single trial.

P(X = k) is the probability of having exactly k failures before n successes.

### Probability Density Function (PDF)

`P(X = k) = ([k+n-1],[k]) * p^n*(1-p)^k`

where `([n], [k])` is the binomial coefficient also called 'n choose k',
`([n], [k]) = (n!)/(k! * (n-k)!)`

### Cumulative distribution function (CDF)

`F(x) = I_p(n,k+1)`

where `I_p` is the regularized incomplete beta function,

## See also

Inverse Negative binomial distribution

Negative binomial distribution Chart

Statistics Calculators