Beta Binomial Distribution Calculator
Calculate overdispersed success-count probabilities when a binomial success probability follows an entered beta distribution.
YOUR CALCULATION
Result
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Calculation steps
Work through one step at a time or reveal the complete calculation. The steps use your current inputs.
How to calculate
Integrate a binomial success probability over the beta mixing distribution using log-gamma normalization.
P(K=k)=choose(n,k) B(k+α,n−k+β)/B(α,β)
Worked example
With α=β=1, every count k=0…n has probability 1/(n+1), unlike the ordinary p=0.5 binomial.
Assumptions and input details
Count k cannot exceed n. This is an entered beta–binomial mixture, not fitted overdispersion or independent resampling of p before each trial. Probabilities may underflow in extreme tails.
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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