Negative Binomial Distribution

Modeling the number of trials needed to achieve a specified number of successes.

A Generalization of the Geometric Distribution

The Negative Binomial distribution answers the question: "How many trials will it take to get my $r$ -th success?" It is a generalization of the Geometric distribution, which is just the special case where $r=1$ .

In finance, a trader might use this to model how many trades it will take to achieve 10 winning trades. A venture capitalist could model how many startups they need to fund to get 3 successful exits.

The Formula

The probability that the

r

-th success occurs on the

k

-th trial is:

P(X=k) = \binom{k-1}{r-1} p^r (1-p)^{k-r}

$k$ is the total number of trials.
$r$ is the desired number of successes.
$p$ is the probability of success on a single trial.

Interactive Negative Binomial Distribution

Adjust the required number of successes (

r

) and the probability (

p

) to see how the distribution changes.

Number of Successes (r): 5

Probability of Success (p): 0.50