Bayes' Theorem Calculator

Free Online Bayes' Theorem Calculator

This Bayes' theorem calculator finds the posterior probability P(A|B) — the updated probability of event A after observing evidence B — from a prior probability and two likelihoods. Enter P(A), P(B|A), and P(B|¬A), and the calculator instantly returns P(A|B), the total probability of B, and both joint probabilities, complete with a probability-tree diagram and a full step-by-step solution.

Whether you're a student solving Bayesian probability problems, a data scientist building a Naive Bayes classifier, or someone interpreting a medical test result, this Bayes' theorem solver handles the entire calculation for you and shows exactly how the answer was derived using the law of total probability.

Bayes' Theorem Formula

Bayes' theorem describes how to update the probability of an event based on new evidence. The formula is:

P(A|B) = [P(B|A) × P(A)] / P(B)

Here, P(A) is the prior probability — what you believed about A before seeing any evidence. P(B|A) is the likelihood — how probable the evidence B is if A is actually true. P(A|B) is the posterior probability — your updated belief about A after observing B. When P(B) isn't given directly, it's calculated using the law of total probability: P(B) = P(B|A)·P(A) + P(B|¬A)·P(¬A).

How to Use This Bayes' Theorem Calculator

Using the calculator takes just three steps. First, enter P(A), the prior probability of the event you're interested in. Second, enter P(B|A), the probability of observing the evidence if A is true. Third, enter P(B|¬A), the probability of observing the same evidence if A is false. The calculator instantly computes P(A|B) — the probability that A is true given that you've observed B — along with a probability-tree diagram showing exactly how the numbers combine.

In the diagram, the tree splits first on whether A happens or not, then on whether B is observed within each branch. Once B is actually observed, only the two cyan-highlighted 'B' branches remain relevant — P(A|B) is simply the top branch's joint probability divided by the sum of both.

Worked Example: The Classic Medical Test Problem

Suppose a disease affects 1% of a population (P(A) = 0.01). A test for the disease is 95% accurate at detecting it when present (P(B|A) = 0.95, the sensitivity), but also gives a false positive 5% of the time on healthy people (P(B|¬A) = 0.05). If someone tests positive, what's the actual probability they have the disease?

P(A ∩ B) = 0.01 × 0.95 = 0.0095. P(¬A ∩ B) = 0.99 × 0.05 = 0.0495. P(B) = 0.0095 + 0.0495 = 0.059. So P(A|B) = 0.0095 / 0.059 ≈ 0.161, or about 16.1% — a surprisingly low number given a '95% accurate' test. This counterintuitive result is exactly why Bayes' theorem is essential: it correctly accounts for how rare the disease is in the first place (the base rate), something an accuracy percentage alone doesn't capture.

Why Base Rates Matter (The Base Rate Fallacy)

The medical test example above illustrates the base rate fallacy — the common mistake of ignoring how rare or common an event is beforehand and focusing only on a test's accuracy. Even a highly accurate test can produce mostly false positives if the condition being tested for is rare, simply because there are so many more healthy people who could trigger a false alarm than sick people who could trigger a true one. Bayes' theorem is the correct mathematical tool for combining accuracy with base rate to get a truthful answer.

Where Bayes' Theorem Is Used

Bayes' theorem is one of the most important tools in probability and statistics, with applications across many fields:

• Students and teachers use it throughout probability, statistics, and machine learning coursework.
• Medical diagnostics rely on it to correctly interpret test results by combining test accuracy with disease prevalence.
• Spam filters and Naive Bayes classifiers use it to calculate the probability an email is spam given the words it contains.
• Legal and forensic reasoning uses Bayesian analysis to weigh evidence like DNA matches against how rare a given genetic profile is.
• Search and recommendation engines use Bayesian updating to refine predictions as new user behavior data comes in.