How this calculator works
The last trial must be a success. The preceding trials contain the other r - 1 successes and all failures, giving the combination factor in the mass function. This tool uses a whole-number success target rather than a general real-valued dispersion parameter. Trials must be independent with constant success chance.
Formula
P(F=k) = C(k+r-1,k)p^r(1-p)^k; E[F] = r(1-p)/p; Var[F] = r(1-p)/p²; T = F + r.
Worked example
For r = 2 and p = 0.5, exactly one failure before the second success has probability 2/8 = 0.25. At most one failure has probability 0.5.
Advertisement
Frequently asked questions
How is this different from a binomial count?
Binomial fixes the number of trials. Negative binomial fixes the number of successes and lets the trial count vary.
Does r = 1 simplify the model?
Yes. It becomes the geometric distribution with the same counting convention.
Can the count be below r?
Total trials below r have probability zero. Failure counts may begin at zero.