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Walk-through

How to Use This Calculator

5 steps

1

Pick Direct or Medical Test

Use the Bayes' Theorem tab when you already know a prior P(A), a likelihood P(B|A), and the total evidence P(B). Use the Medical Test tab when you have a test's sensitivity, specificity, and the condition's prevalence — the calculator derives the evidence for you.

2

Enter your probabilities

Type each value as a percentage between 0 and 100. The result updates instantly as you type, and two example-scenario chips on each tab load a ready-made dataset if you want to start from a known case.

3

Read the posterior

The hero number is P(A|B), the posterior probability — how likely A is now that you've observed B. The detail line below it spells out the arithmetic (Direct tab) or the false-positive share (Medical tab).

4

Compare prevalence scenarios

On the Medical Test tab, the scenario-comparison card holds your sensitivity and specificity fixed and shows the posterior at a rare (1%) versus a common (20%) prevalence, so you can see exactly how much the base rate moves the answer.

5

Read the Explanation tab

The Explanation tab walks through what Bayes' theorem means, why the base rate matters more than most people expect, and a fully worked rare-disease example, with a live summary of your own last calculation at the bottom.

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Reference

Formula & Methodology

2 formulas

Bayes' theorem

P(A|B) = P(B|A) · P(A) / P(B)

The posterior P(A|B) equals the likelihood P(B|A) times the prior P(A), divided by the evidence P(B) — the total probability of observing B across every way it can happen. All four terms are probabilities between 0 and 1 (0% and 100%).

Evidence from a medical test

P(B) = sensitivity · prevalence + (1 − specificity) · (1 − prevalence)

When B is 'the test came back positive,' its total probability is the sum of two paths: true positives (sensitivity times how common the condition is) plus false positives (the false-positive rate times how uncommon the condition is). Plugging this P(B) into Bayes' theorem gives the positive predictive value (PPV).

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Glossary

Key Terms Explained

7 terms

Bayes' theorem A rule for updating the probability of a hypothesis A after observing evidence B: P(A|B) = P(B|A)·P(A) / P(B). It formalizes how new information should change a prior belief.

Posterior P(A|B) — the updated probability of A after taking the evidence B into account. This is the number the calculator returns as its headline result.

Prior P(A) — the probability assigned to A before seeing any evidence. In the medical-test framing, the prior is the condition's prevalence in the population being tested.

Likelihood P(B|A) — the probability of observing the evidence B assuming the hypothesis A is true. In a medical test, this is the test's sensitivity: how often it correctly flags someone who has the condition.

Evidence P(B) — the total, unconditional probability of observing B, summed across every way B could have happened (both when A is true and when it isn't). Also called the marginal likelihood or normalizing constant.

Conditional probability The probability of one event given that another has occurred, written P(X|Y). Bayes' theorem is a rule for relating P(A|B) to P(B|A) — the two conditional probabilities are not the same thing, which is the root of most Bayesian reasoning mistakes.

Base rate How common a condition or event is in the overall population before any test or evidence is considered — equivalent to the prior. Ignoring the base rate and over-weighting a single piece of evidence is the base-rate fallacy.

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Scenarios

Real-World Examples

3 worked examples

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Direct form

Weather forecast

Prior P(A) 30% Likelihood P(B|A) 90% Evidence P(B) 45%

There's a 30% base chance of rain today, and clouds appear 90% of the time it rains. Clouds appear 45% of all days, rain or not. Plugging in: P(rain|clouds) = 0.9 × 0.3 / 0.45 = 0.6, so seeing clouds this morning raises the chance of rain from 30% to 60% — the evidence roughly doubled the prior.

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Medical test

Rare-disease screening

Sensitivity 90% Specificity 90% Prevalence 1%

A test that's 90% sensitive and 90% specific sounds highly reliable, but for a disease only 1% of people have, a positive result still means just an 8.3% chance of actually being sick. Out of 10,000 people, the test flags 1,080 positive — but only 90 of them truly have the disease, because there are far more healthy people (9,900) generating false alarms than sick people generating true ones. This is the classic base-rate fallacy.

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Same test, different prior

High-risk vs. general population

Sensitivity 90% Specificity 90% Prevalence (high-risk group) 20%

Run the exact same test on a group where the condition is 20 times more common (20% prevalence instead of 1%) and the posterior jumps from 8.3% to 69.2% — the test itself didn't change at all. Only the prior did. This is why the same screening test is interpreted very differently for a high-risk group than for the general population.

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Deep Dive

Understanding Bayes' Theorem

3 chapters · 4 min read

Bayes' theorem is the mathematical rule for updating a belief in light of new evidence. It underlies medical testing, spam filters, legal reasoning, and everyday judgment calls — and it's also the source of one of the most common probability mistakes people make: trusting a piece of evidence without accounting for how rare the thing being tested for actually is.

How the calculator works

The Bayes' Theorem tab applies P(A|B) = P(B|A)·P(A) / P(B) directly to whatever prior, likelihood, and evidence you enter. The Medical Test tab is the same formula wearing different labels: sensitivity is the likelihood P(B|A), prevalence is the prior P(A), and the evidence P(B) — the overall chance of testing positive — is derived automatically from sensitivity, specificity, and prevalence, since you rarely know it directly for a real test.

Why the base rate matters more than the test

A test's accuracy (sensitivity and specificity) tells you how good the test is at spotting true cases and true non-cases. But the posterior — what a positive result actually means for you — depends just as much on the prevalence, or base rate. When a condition is rare, the pool of healthy people is enormous compared to the pool of sick people, so even a small false-positive rate produces more false alarms than true detections. Ignoring this is the base-rate fallacy, and it's why doctors, forecasters, and fraud analysts are trained to always ask 'how common is this to begin with?' before trusting a single positive signal.

Direct vs. medical framing — same formula, two names

The Direct tab and the Medical Test tab compute the identical Bayes' theorem calculation; they just use domain-specific names for the same three ingredients. If you switch a scenario's numbers between the two tabs consistently (prior = prevalence, likelihood = sensitivity, evidence = the derived positive-test rate), you'll get the same posterior either way — a useful sanity check for understanding the formula, and the reason the calculator's default values on both tabs tell the same rare-disease story.

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Questions

Frequently Asked Questions

6 questions

What is the formula for Bayes' theorem?+

P(A|B) = P(B|A) · P(A) / P(B). The posterior P(A|B) equals the likelihood P(B|A) times the prior P(A), divided by the evidence P(B). All four terms are probabilities between 0% and 100%.

How do I use this for a medical test?+

Switch to the Medical Test tab and enter the test's sensitivity (true-positive rate), specificity (true-negative rate), and the condition's prevalence in the population. The calculator derives the evidence P(B) automatically and returns the positive predictive value (PPV) — the chance a positive result is a real case.

What is the base-rate fallacy?+

It's the common mistake of judging a positive test result purely by the test's accuracy while ignoring how rare the condition is. A highly accurate test can still produce mostly false positives when the condition it's testing for is uncommon, because the healthy population vastly outnumbers the sick one.

What is a prior?+

The prior, P(A), is your starting probability for a hypothesis before you see any evidence. In the medical-test framing it's the prevalence — how common the condition is in the population being tested — before anyone takes the test.

What units does this calculator use?+

Every input and output is a probability, entered and displayed as a percentage between 0% and 100%. There are no currency, distance, or weight conversions involved.

What is PPV, and how does it relate to Bayes' theorem?+

PPV (positive predictive value) is exactly the posterior P(A|B) computed for the medical-test framing — the probability that someone actually has the condition given that they tested positive. It's the same Bayes' theorem calculation as the Direct tab, just applied to sensitivity, specificity, and prevalence instead of an abstract prior, likelihood, and evidence.

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Bayes' Theorem Calculator

Direct Form

Medical Test

Same test, different prevalence

What Bayes' theorem says

Why the base rate matters

Walk through the rare-disease example

Prior vs. posterior — updating your belief

How to Use This Calculator

Pick Direct or Medical Test

Enter your probabilities

Read the posterior

Compare prevalence scenarios

Read the Explanation tab

Formula & Methodology

Key Terms Explained

Real-World Examples

Direct form

Medical test

Same test, different prior

Understanding Bayes' Theorem

How the calculator works

Why the base rate matters more than the test

Direct vs. medical framing — same formula, two names

Frequently Asked Questions

Get a branded PDF of your results

Bayes' Theorem Calculator

Direct Form

Medical Test

Same test, different prevalence

What Bayes' theorem says

Why the base rate matters

Walk through the rare-disease example

Prior vs. posterior — updating your belief

How to Use This Calculator

Pick Direct or Medical Test

Enter your probabilities

Read the posterior

Compare prevalence scenarios

Read the Explanation tab

Formula & Methodology

Key Terms Explained

Real-World Examples

Understanding Bayes' Theorem

How the calculator works

Why the base rate matters more than the test

Direct vs. medical framing — same formula, two names

Frequently Asked Questions

Get a branded PDF of your results

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