How this calculation works
Uncertainty about a success probability affects predictions for future counts. A beta prior combines with observed Bernoulli trials to give a beta posterior, which can then be integrated into a beta-binomial predictive distribution. This page reports the probability of at least an entered number of future successes and shows the complete bounded predictive count table. It does not substitute the posterior mean into an ordinary binomial distribution.
Posterior A = a+s, B = b+n−s; P(K=k) = choose(m,k) Beta(A+k,B+m−k)/Beta(A,B)A worked example
With a uniform Beta(1,1) prior and no observations, the predictive counts for four future trials are zero through four, each with probability 0.2. The probability of at least three successes is therefore 0.4. This uniform predictive count example provides a direct check on the beta-function calculation without relying on an engine snapshot.
Good to know
The model assumes exchangeable independent Bernoulli trials conditional on one fixed probability. Time trends, clustered outcomes, changing treatment and inconsistent success definitions invalidate that simple pooling. No automatic prior selection, model checking or causal conclusion is provided. Future counts are capped at five hundred to keep the complete probability table bounded and usable.
Reference: NIST — beta conjugacy and Bayesian reliability (opens in a new tab) ↗
What this calculation answers
Uncertainty about a success probability affects predictions for future counts. A beta prior combines with observed Bernoulli trials to give a beta posterior, which can then be integrated into a beta-binomial predictive distribution.
This page reports the probability of at least an entered number of future successes and shows the complete bounded predictive count table. It does not substitute the posterior mean into an ordinary binomial distribution.
The Binomial probability calculator uses a fixed probability. The posterior predictive count here instead integrates uncertainty about that probability.
Use the right input conventions
Prior alpha and beta must be positive. Successes cannot exceed observed trials, and the predictive threshold cannot exceed the number of future trials. A zero observed-trial count leaves the prior unchanged.
All observations are assumed to share the same unknown success probability. Prior shapes are user assumptions, not automatically estimated weights or a claim that every prior is appropriate for the task.
Interpret the result
The posterior mean is a point summary, while the predictive distribution carries probability uncertainty into future counts. This can make the predictive count more dispersed than a binomial with the same mean.
A posterior probability is conditional on the chosen prior and likelihood. It is not a frequentist confidence level or a guarantee about the next group of trials.
Work through the example
With a uniform Beta(1,1) prior and no observations, the predictive counts for four future trials are zero through four, each with probability 0.2.
The probability of at least three successes is therefore 0.4. This uniform predictive count example provides a direct check on the beta-function calculation without relying on an engine snapshot.
Where the model stops
The model assumes exchangeable independent Bernoulli trials conditional on one fixed probability. Time trends, clustered outcomes, changing treatment and inconsistent success definitions invalidate that simple pooling.
No automatic prior selection, model checking or causal conclusion is provided. Future counts are capped at five hundred to keep the complete probability table bounded and usable.
A few questions, answered
Why not use a binomial with the posterior mean?
That plugs in one probability and omits the posterior uncertainty integrated by the beta-binomial prediction.
What changes in the comparison?
Update the same prior with eight successes in ten trials before predicting another four.