Negative binomial distribution calculator
Find the probability of failures before the r-th success or total trials to that success. See the CDF, mean and variance.
How to use this page
- Load an example or enter your own data.
- Choose the calculation type and units where available.
- Read the result, additional measures and substituted formula.
What this page does not do: Trials must be independent with a constant p greater than zero. r: 1–1000, x: 0–10,000. Very small probabilities may round to zero. This model does not predict the next lottery draw.
Further reading
Formula
P(K=k) = C(k+r−1,k) pʳ (1−p)ᵏ Repeat independent trials with a constant success probability p until r successes occur. K counts failures before the last success. Total trials T = K + r includes successes. The last trial must be a success.
For r = 3, p = 0.5 and k = 2, C(4,2) × 0.5³ × 0.5² = 0.1875, or 18.75%. The probability of at most two failures is 50%. Selecting five total trials gives the same point probability.
A binomial model fixes the number of trials and counts successes. A negative binomial model fixes successes and counts the trials or failures needed. For r = 1 it is the geometric distribution.
Frequently asked questions
How can I check the example?
r = 3, p = 0.5, k = 2: exactly 18.75%, up to two failures 50%.
What should I check before interpreting the result?
Trials must be independent with a constant p greater than zero. r: 1–1000, x: 0–10,000. Very small probabilities may round to zero. This model does not predict the next lottery draw.
See also
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