This tool computes a posterior for a yes or no hypothesis. You give a prior $P(H)$, a likelihood $P(E\mid H)$, and the chance of the same evidence when the hypothesis is false, $P(E\mid \text{not }H)$. It reports
$$P(H\mid E)=\frac{P(E\mid H)\,P(H)}{P(E)}$$
with
$$P(E)=P(E\mid H)\,P(H)+P(E\mid \text{not }H)\,P(\text{not }H)$$
and $P(\text{not }H)=1-P(H)$. The large number on the page is $P(H\mid E)$. The lines under it are the same formula with your inputs plugged in, including the two pieces that add up to $P(E)$.
The number is meaningful when $H$ and its opposite are the only two possibilities you are willing to consider, when the three inputs are probabilities in that same model, and when $E$ is the evidence you actually observed. A classroom test, a spam flag, or a card draw can fit once you name those events. A binomial count is a different job. That tool fixes a success probability and asks for the chance of exactly $k$ successes. Here those rates are inputs, and the output is an updated belief.
This is one update for one binary hypothesis. It does not estimate a prior from a data file, and it does not check that a later observation is independent of this one. Name the events in the Bayes theorem explainer before you trust a digit.
How to use the two input styles
Probability style is the formula as written. Enter $P(H)$, $P(E\mid H)$, and $P(E\mid \text{not }H)$, each in $[0,1]$, then press Compute. The defaults are a rare hypothesis: prior $0.01$, likelihood $0.99$, and a $0.05$ chance of the evidence when $H$ is false. That is the classroom test from the Bayes explainer, written as probabilities instead of a hall of people.
Natural frequency style replaces the prior with a head count. Enter a population size $N$, how many of those $N$ have $H$, and the same two likelihoods. The page sets $P(H)$ equal to the count divided by $N$, then runs the same formula. It also prints the expected true positive count and the expected false positive count, so you can see which group is supplying the evidence. The defaults are $N=10000$ and $100$ people with $H$, which is the same prior $0.01$. Those counts are expectations. If the rates do not land on whole people, the page still uses the exact ratio for the posterior.
Type probabilities as decimals. $0.99$ means 99 percent, and $99$ is rejected. Switching to counts rounds $P(H)\times N$ to a whole person. A repeating prior can move by about one person, so keep an exact decimal in probability style.
How to read the output
The large number is $P(H\mid E)$. When it is not a short decimal, it is rounded to 6 places and the word “about” appears with a percent. The lines underneath keep more digits. They are the same formula, not a second model.
$P(\text{not }H)$ is one minus the prior. The next two lines are the pieces before you divide: evidence from $H$, and evidence from the opposite. Their sum is $P(E)$. Dividing the piece from $H$ by that sum is the posterior. The other piece over the same sum is $P(\text{not }H\mid E)$. Those two posteriors add to $1$.
On the defaults, the piece from $H$ is $0.0099$ and the piece from the opposite is $0.0495$. So $P(E)=0.0594$ and
$$P(H\mid E)=\frac{0.0099}{0.0594}=\frac{1}{6}\approx 0.166667.$$
A bit under 17 percent, not $0.99$. Most of that evidence is false positives from the large group without $H$. Quoting the likelihood as if it were the posterior is base rate neglect. Count mode with $10000$ people prints $99$ expected true positives and $495$ expected false positives, and $99/(99+495)$ is the same $1/6$. If both likelihoods are positive and the prior sits strictly between $0$ and $1$, the page also prints the likelihood ratio and the prior and posterior odds. That is the same fraction in another form. If $P(E\mid H)$ is smaller than $P(E\mid \text{not }H)$, the posterior falls below the prior. That is a real update, not a broken input.
A second check you can do by hand
Two jars, from the probability and statistics topic. The prior on jar 1 is $1/2$. Jar 1 produces red with probability $3/4$, and jar 2 with probability $1/4$. Enter prior $0.5$, likelihood $0.75$, and $0.25$ for $P(E\mid \text{not }H)$. The posterior for jar 1 given red is $0.75$. The match with the likelihood is an accident of these counts. Change the prior to $0.1$ and leave the likelihoods alone. The posterior drops to $0.25$. If you still get $0.75$, you ignored the prior.
Common pitfalls
$P(E\mid H)$ and $P(H\mid E)$ are different questions. Sensitivity is about the evidence given the hypothesis. The posterior is about the hypothesis given the evidence. The labels stay in one order. The button will not swap a story you told backwards.
A probability outside $[0,1]$ is rejected, and so is a head count larger than the population, a population below $1$, or a blank box. If $P(E)=0$, the posterior is undefined and the page stops instead of dividing by zero. A prior of $0$ stays $0$ when the evidence can still come from the opposite hypothesis. A prior of $1$ stays $1$ when the evidence can come from $H$. Population size stops at one billion.
The tool is binary: one hypothesis and its opposite. The three box card problem in the explainer has three priors, so do not fold those boxes into a single $H$ until you have decided which event $H$ is. A second observation needs its own likelihood. Feed the posterior back in as the next prior only when the new evidence is independent of the old evidence given $H$. The explainer shows a card draw where that independence fails.
Related reading
Read the story behind the defaults in Bayes theorem without the fog. The surrounding map is probability and statistics. Short definitions sit at the glossary entries for Bayes theorem, conditional probability, and probability.
A fixed number of independent trials with a constant success probability belongs on the binomial probability calculator. Counting selections, without a probability model, is the combinatorics calculator. A continuous tail is the normal CDF and z score calculator. None of those three updates a prior. This page does, and only for the binary hypothesis you type.