Negative Binomial Probability Calculator
Calculate a failure-count PMF and inclusive tails when independent Bernoulli trials stop at a specified number of successes.
YOUR CALCULATION
Result
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Calculation steps
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How to calculate
Use the failures-before-rth-success convention with log-gamma PMFs and regularized-beta probability tails.
P(K=k)=C(k+r−1,k)p^r(1−p)^k; E[K]=r(1−p)/p
Worked example
With r=3 successes, p=0.5 and k=2 failures, exactly five trials are needed for that outcome. P(K=2)=0.1875, P(K≤2)=0.5 and P(K≥2)=0.6875.
Assumptions and input details
Required successes 1–1000, failures 0–100000 and success probability from 0.000001% to 100%. Trials are independent and share the same p. This counts failures, not total trials; total trials equal failures plus required successes. Both tails include k and therefore do not sum to one. Extreme tiny probabilities can underflow; no independence or real-world success-rate estimation is performed.
References: SciPy: negative-binomial failures convention
Using this calculator
Can I enter my own problem?
Yes. Replace the example values with your own inputs. The result, calculation steps, tables and applicable diagrams update from those inputs. Blank or invalid values clear the result; example values are never silently substituted for missing inputs.
Are the results exact?
Results use floating-point arithmetic and displayed rounding. Read the formula, assumptions and any stated data precision before interpreting the result.
Is my input uploaded?
The calculation runs in your browser. No account, file upload or external AI service is required. Inputs are not saved between page reloads.
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