A probability distribution over a probability — and the answer to “5 out of 5”.
A distribution over a proportion
P(p ≥ 0.5000)
0.886719
88.672% — under Beta(8.0000, 4.0000)
7/10 observed. The plain proportion says 0.7000; this says 0.666667. The difference is shrinkage toward a half, and it shrinks as evidence accumulates: it is worth 0.033333 at n = 10, and would be roughly 0.003922 with ten times the data at the same rate.
Mean
0.666667
Mode
0.700000
Standard deviation
0.130744
95% interval
0.3903 – 0.8907
Here the mode sits above the mean, by 0.033333. Which way round depends on the skew, and a beta can go either way: with α > β the tail runs left and the mode is the larger of the two; with α < β it reverses; at α = β they coincide exactly. That is worth stating because the neighbouring lognormal has a fixed order — mode < median < mean, always — and it is easy to assume every skewed distribution behaves that way.
What this tool shows
Five successes out of five does not mean 100%. A plain proportion says it does, which would mean no future trial can ever fail — on five observations. The beta gives 6/7 = 0.857: high, not certain. Enter successes and failures and this does that update for you.
Probabilities and quantiles on any Beta(α, β)
The conjugate update from raw successes and failures
Laplace’s rule of succession, applied automatically
Beta(8,4) has mode 0.700 above mean 0.667 — the order is not fixed.
Updated 8 September 2026 · Works in any browser, no installation
The beta is a distribution over a proportion. Where a binomial takes p as known and gives you counts, the beta takes counts as known and gives you a distribution over p — the same relationship read backwards, which is why the two fit together so neatly.
At a glance
Formula shown
f(p) ∝ p^(α−1)(1 − p)^(β−1) on [0, 1]. Mean α/(α + β); mode (α − 1)/(α + β − 2) when both exceed 1. Starting from Beta(α₀, β₀) and observing s successes and f failures, the posterior is Beta(α₀ + s, β₀ + f) — the update is addition. From a uniform Beta(1,1) prior the posterior mean is (s + 1)/(n + 2).
Scenario support
Conversion rates from small samples; click-through and A/B tests; batting averages and shooting percentages; defect and pass rates; any "x out of n" where n is small enough that the plain proportion is unstable; a prior over an unknown probability.
Educational estimate
Planning support from the values you enter — not professional advice.
Five out of five, and the answer that is not 100%
A new process passes five tests out of five. What is the probability the sixth passes?
The plain proportion says 5/5 = 100%, and taken literally that asserts failure is impossible — a claim about all future trials, from five observations. Nobody believes it, which is why people quietly avoid quoting it rather than fixing it.
The beta fixes it by counting rather than dividing. Start from Beta(1, 1) — the flat distribution, every proportion equally plausible. Add the successes to α and the failures to β. Five and zero gives Beta(6, 1), whose mean is 6/7 = 0.857. High, not certain, and it moves toward 1 as evidence accumulates rather than jumping there.
This is Laplace’s rule of succession, and the whole of it is (s + 1)/(n + 2). Laplace derived it in the 1770s asking what odds to give on the sun rising tomorrow given that it has always risen — his point being that “always so far” is not the same as certainty, and that the difference can be quantified.
The same correction runs the other way and matters more. Zero adverse events in 30 patients does not mean the rate is zero — it means the rate is probably below about 10%, and a single trial in 31 would be unremarkable. The clinical shorthand is the rule of three: with no events in n trials, the upper 95% bound is roughly 3/n. Drug safety monitoring is built on this, because “we saw none” is otherwise a dangerously confident thing to write down.
The correction is largest exactly where it is most needed. At n = 5 it moves the answer by 0.14; at n = 1000 by less than 0.001. It is not a thumb on the scale, it is a small-sample repair that switches itself off.
Why the update is just addition
The beta is the conjugate prior for a binomial, which sounds technical and means something very practical: start with a beta, observe binomial data, and the answer is another beta. No integration, no simulation, no approximation — you add the successes to α and the failures to β.
The algebra is short enough to see. A beta density is proportional to pα−1(1 − p)β−1. A binomial likelihood is proportional to ps(1 − p)f. Multiply them and the exponents add. That is the whole mechanism.
The practical consequence is that α and β are readable as counts. Beta(30, 10) is “something like 29 successes and 9 failures’ worth of belief”. That makes priors arguable in plain language: someone who says the conversion rate is around 3% with the confidence of about 200 observations is specifying Beta(6, 194), and you can push back on the number of observations rather than on an abstraction.
α + β is the sample size the prior is worth, and it tells you when data will overwhelm it. A Beta(2, 2) prior is worth four observations and vanishes immediately. A Beta(200, 800) prior is worth a thousand and will not move for a small test — which is a legitimate choice if you genuinely have that much history, and a way of ignoring your own experiment if you do not.
Updating is order-independent and incremental. Ten successes now, or one a day for ten days, or all at once at the end — the posterior is identical. This is what makes the beta the natural engine for online A/B testing and multi-armed bandits: state is two numbers per variant, updated in constant time.
Reading the shape from α and β
Two parameters produce a striking variety of shapes on [0, 1], and each corresponds to a different state of knowledge.
α = β = 1 is the uniform distribution — exactly flat, every proportion equally plausible. The uniform on [0, 1] is a beta, and this is the “before any data” state.
Both above 1 gives a single interior peak at (α − 1)/(α + β − 2), narrowing as the parameters grow. This is the ordinary case after real data.
Both below 1 gives a U-shape, with the density rising without bound at both ends. It is genuinely infinite there and still integrates to 1 — unbounded density is not a bug, it is what a density is allowed to do. Beta(½, ½) is the Jeffreys prior, and it encodes a belief that the proportion is near one extreme without saying which.
Mode and mean differ, and the order is not fixed. Beta(8, 4) has mean 0.667 and mode 0.700 — the mode is the larger. Reverse the parameters and it reverses. At α = β they coincide. This is worth stating explicitly because a lognormal has a fixed order — mode < median < mean, always — and it is a natural mistake to assume every skewed distribution behaves the same way. It does not.
A large α + β means a narrow distribution. Beta(3, 2) and Beta(300, 200) have the same mean of 0.6 and completely different widths — the second is confident, the first barely committed. The mean alone never tells you which you have.
Credible intervals, and what they are not
The 95% interval this tool reports is a credible interval, and it means the thing people usually assume a confidence interval means.
A credible interval says the parameter has a 95% probability of lying inside it, given your prior and data. That is a direct statement about the unknown proportion, and it is the sentence almost everyone reaches for when reading a confidence interval — where it is not correct, because a frequentist interval is a statement about the procedure’s long-run coverage rather than about this particular parameter.
The distinction is philosophical until the sample is small, at which point it becomes arithmetic. For 0 successes in 20 trials the ordinary Wald interval collapses to a single point at zero — it reports no uncertainty at all about an event it never observed. The beta interval runs from about 0 to 0.16, which is the honest answer.
Two intervals can be quoted from a beta and they are not the same. The equal-tailed interval cuts 2.5% from each end and is what this tool reports; the highest-density interval takes the shortest interval containing 95% of the mass. For a symmetric beta they coincide; for a skewed one the highest-density interval is narrower and can touch 0 or 1, which an equal-tailed one never does. Neither is wrong — but two tools reporting “the 95% interval” can legitimately disagree, and this is usually why.
The interval also inherits the prior. With very little data it is mostly a description of your assumption rather than of the evidence, and quoting it as a finding overstates what was observed. Report the prior alongside it whenever n is small.
Where it earns its place
Any bounded proportion estimated from a countable number of trials.
Conversion and click-through rates. The canonical case, and the one where the plain proportion fails most visibly: 2 conversions from 8 visitors is not a 25% conversion rate in any useful sense, and the beta says so by producing an interval wide enough to be honest.
A/B testing and bandits. Keep a beta per variant and the probability that B beats A is a direct calculation rather than a hypothesis test. Thompson sampling — draw once from each variant’s beta and serve the winner — is a complete adaptive algorithm in one line.
Sports and performance rates. Batting averages, free-throw percentages, save rates. The shrinkage the beta applies is exactly why early-season leaders regress: a 0.450 average over 20 at-bats is mostly noise, and the posterior says so without needing a separate argument.
Quality and safety rates. Defect rates, pass rates, adverse event rates — particularly when the observed count is zero and the plain proportion is useless.
Where it does not fit: anything unbounded (use a lognormal or exponential), counts rather than proportions (a Poisson or negative binomial), and trials that are not independent — correlated trials make the posterior far too confident, and no choice of α and β repairs that.
Sources and methodology
References for the conjugate update and the small-sample rules.
Method. Tail probabilities use the regularised incomplete beta function, which is exact rather than simulated. Quantiles invert it by bisection rather than by Newton’s method, deliberately: when α or β is below 1 the density is unbounded at an endpoint, so the derivative Newton divides by runs to infinity and the iteration can diverge — bisection cannot, on a monotone function, and 200 halvings of [0, 1] reach the limit of double precision. The suite asserts that the inverse round-trips through the CDF at Beta(½, ½) as well as at well-behaved parameters, that Beta(1, 1) is the uniform to twelve decimals, and that Beta(8, 4) has its mode above its mean — the case that stops this page claiming a fixed mean-mode order. That engine is verified on every change against 219 assertions. The count and the per-case breakdown are published on the formula verification page.
Related calculators
Where this goes next:
Uniform DistributionHandles the continuous and discrete uniform as the separate distributions they are — P(X = x) is zero in one and 1/n in the other, and the variances differ for identical endpoints.
Binomial DistributionExact binomial probabilities at any n — including thousands, where a factorial overflows — with the normal approximation beside them and its error measured, which is 0.6% at the centre and 261% in the tail.
Confidence IntervalIntervals for a mean or a proportion using t at every sample size and Wilson rather than the textbook Wald formula — with both methods shown, because Wald returns [0,0] at zero successes.
Lognormal DistributionTakes log-scale parameters or your data's own mean and SD and converts between them — because μ describes ln x, not x, and entering the wrong one is wrong by orders of magnitude with nothing to flag it.
ProbabilityTwo events, repeated trials and Bayes, with the three usual errors handled — the dropped overlap in P(A or B), n×p instead of the complement, and the base rate that makes a 99% test 17% right.
Sample SizeResponses needed for a target margin of error, with the finite-population correction and a table of the whole cost curve — because n scales with 1/margin², so the last point of precision costs more than the first ten.
An educational tool. Results assume independent trials with a constant underlying probability; correlated trials make the posterior far more confident than the evidence supports, and no choice of α and β corrects for that. With small samples the interval reflects the prior as much as the data.
Published a beta calculator that takes raw successes and failures and runs the conjugate update, so five successes out of five returns 6/7 rather than the 100% a plain proportion claims — Laplace's rule of succession, which is just (s + 1)/(n + 2).
Handles the zero-events case the same way, where the plain proportion returns an impossible 0% and the counted answer is small but not ruled out. That is the arithmetic behind the clinical rule of three.
Inverts the CDF by bisection rather than Newton's method, because the density is unbounded at an endpoint whenever a parameter falls below 1 and the derivative Newton divides by runs to infinity there.
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