What is Bayes' Theorem?

Bayes' Theorem is a fundamental formula in probability theory that allows us to update our beliefs based on new evidence. It calculates the probability of an event occurring given that another event has already occurred.

The formula is: P(A|B) = [P(B|A) × P(A)] / P(B)

Where:

P(A|B) = Posterior probability (what we want to find)
P(B|A) = Likelihood (probability of evidence given hypothesis)
P(A) = Prior probability (initial belief)
P(B) = Marginal probability (total probability of evidence)

Example: A disease test with 0.1% prevalence and 5% false positive rate shows that even with a positive test result, the actual probability of having the disease is only about 1.96% due to the high number of false positives.

Bayes' Theorem Calculator

Prior Probability P(H) - Probability of Hypothesis

Enter as decimal (e.g., 0.001 for 0.1%)

Likelihood P(E|H) - Probability of Evidence given Hypothesis is True

Sensitivity / True Positive Rate (e.g., 1.0 for 100%)

False Positive Rate P(E|¬H) - Probability of Evidence given Hypothesis is False

Enter as decimal (e.g., 0.05 for 5%)

Results

Posterior Probability P(H|E):

Probability of Evidence P(E):

True Positives:

False Positives:

Bayes' Theorem Calculator

What is Bayes' Theorem?

Bayes' Theorem Calculator

Results

Posterior Probability Trend

Calculation History