Conditional Probability and Bayes' Theorem Calculator

Last updated: 2026-05-20

Conditional probability

Conditional probability is the probability that event A occurs given that event B is known to have occurred. Written P(A|B), it narrows the sample space from all possible outcomes to only those in which B is true, then measures the fraction of those that also involve A. This calculator takes three inputs — the prior probability P(A), the likelihood P(B|A), and the false-positive rate P(B|¬A) — and computes the posterior probability P(A|B) via Bayes' theorem, along with the joint probability P(A∩B) and the total probability P(B).

Bayes' theorem step by step

Bayes' theorem connects the posterior to the prior through two pieces of evidence:

Step 1 — Joint probability

$P(A \cap B) = P(A) \cdot P(B|A)$

This is the probability that both A is true and evidence B appears.

Step 2 — Total probability of B

$P(B) = P(A) \cdot P(B|A) + (1 - P(A)) \cdot P(B|\lnot A)$

B can occur via two paths: when A is true (weighted by the prior) and when A is false (weighted by the false-positive rate). Summing both paths gives the overall chance of seeing B.

Step 3 — Posterior via Bayes' theorem

$P(A|B) = \frac{P(A \cap B)}{P(B)} = \frac{P(A) \cdot P(B|A)}{P(A) \cdot P(B|A) + (1-P(A)) \cdot P(B|\lnot A)}$

The posterior is the fraction of "B events" that also involve A.

Why the formula works

Among all the occasions when B occurs, some are true positives (A was true and produced B) and some are false positives (A was false but B appeared anyway). Bayes' theorem counts both groups precisely. When the false-positive pool is large — either because P(B|¬A) is high or because ¬A is common — it dilutes the true positives and the posterior drops.

Worked example: medical diagnosis

A disease affects 1% of the population. A diagnostic test has 99% sensitivity (P(B|A) = 0.99) and a 5% false-positive rate (P(B|¬A) = 0.05). After a positive test result, the probability that the person actually has the disease is computed as follows.

Step 1: P(A∩B) = 0.01 × 0.99 = 0.0099

Step 2: P(B) = 0.01 × 0.99 + 0.99 × 0.05 = 0.0099 + 0.0495 = 0.0594

Step 3: P(A|B) = 0.0099 / 0.0594 ≈ 16.7%

Despite the test being 99% sensitive, a positive result corresponds to only roughly a 1-in-6 chance of disease. The healthy 99% of the population generates far more false positives (≈5%) than the sick 1% generates true positives (≈1%), so the false positives dominate. This is why confirmatory testing is standard medical practice.

The base rate fallacy

The base rate fallacy is the error of ignoring the prior probability when evaluating evidence. In the example above, focusing only on the test's 99% accuracy ignores the prior information that the disease is rare. The rarer the event, the lower the false-positive rate must be before a positive test becomes reliable. Decreasing P(A) in the calculator lowers the posterior even when the test accuracy is high.

Independence

If P(B|A) equals P(B|¬A), evidence B is statistically independent of A, and observing B gives no information about whether A occurred. In that case the total probability P(B) equals the common value of both likelihoods, and Bayes' theorem simplifies to P(A|B) = P(A). Setting P(B|A) = P(B|¬A) = 50% in the calculator makes the posterior equal the prior regardless of the value of P(A).