Calculator inputs
Use a Beta prior and binomial evidence to estimate an unknown success probability. Results appear above this form after submission.
Formula used
θ ~ Beta(α, β)
s successes and f failures, where n = s + f
θ | data ~ Beta(α + s, β + f)
E[θ | data] = (α + s) / (α + β + s + f)
Mode = (α + s - 1) / (α + β + s + f - 2), when both updated parameters exceed 1
Var(θ | data) = [(α + s)(β + f)] / [((α + β + s + f)2)(α + β + s + f + 1)]
P(next success | data) = Posterior mean
How to use this calculator
- Enter prior alpha and beta to express your starting belief about the unknown success probability.
- Add the observed counts for successes and failures from your sample or experiment.
- Choose a credible interval level such as 95% for uncertainty reporting.
- Enter a threshold if you want the probability that the true rate exceeds a target.
- Submit the form to update the prior into a posterior distribution.
- Review the summary, detailed table, graph, and predictive probability.
- Use the CSV or PDF buttons to export the current result set.
Example data table
Illustrative scenarios using a 95% interval and threshold 0.70.
| Scenario | Prior α | Prior β | Successes | Failures | Posterior Mean | 95% Credible Interval | P(θ > 0.70) |
|---|---|---|---|---|---|---|---|
| Website conversion test | 2.0 | 2.0 | 18 | 7 | 0.6897 | 0.5133 to 0.8412 | 47.24% |
| Defect-free production rate | 5.0 | 2.0 | 36 | 9 | 0.7885 | 0.6688 to 0.8871 | 93.27% |
| Survey agreement rate | 1.5 | 3.0 | 12 | 18 | 0.3913 | 0.2379 to 0.5567 | 0.01% |
Frequently asked questions
1. What does this calculator estimate?
It estimates an unknown success probability after combining prior belief with observed successes and failures. The output includes posterior mean, credible interval, posterior spread, threshold probability, and a predictive chance for the next success.
2. Why use alpha and beta as prior inputs?
Alpha and beta define a Beta distribution, a common prior for probabilities between zero and one. They act like prior evidence, where larger values create stronger prior influence and smaller values allow new data to dominate faster.
3. What is the difference between prior mean and posterior mean?
The prior mean reflects your belief before observing data. The posterior mean updates that belief after combining the prior with the observed sample. It is the main Bayesian point estimate returned by this calculator.
4. What is a credible interval?
A credible interval gives the range where the parameter is likely to lie under the posterior distribution. Unlike a frequentist confidence interval, it directly expresses posterior uncertainty about the parameter after observing the data.
5. When is the posterior mode useful?
The posterior mode is useful when you want the most likely parameter value under the posterior density. It can differ from the posterior mean, especially with small samples or strongly skewed priors.
6. What does P(θ > threshold) mean?
It is the posterior probability that the true success rate exceeds a target you define. This is especially useful in decision-making, screening tests, quality control, and product performance thresholds.
7. Can I use this for A/B tests or quality control?
Yes. It works well for any binary outcome setting, such as conversions, pass-fail checks, defect-free units, approvals, retention events, and similar success-versus-failure measurements.
8. Why does the prior weight decrease with larger samples?
As more observations arrive, the data provides stronger evidence than the initial prior. That causes the posterior estimate to rely less on the prior and more on the observed sample proportion.