Bayes' Theorem

Bayes' Theorem tells you how to reverse conditional probabilities: it turns a known probability of evidence given a hypothesis into the probability of the hypothesis given the evidence. It works because the same joint event can be written two ways, so P(A and B) = P(B|A)P(A) = P(A|B)P(B), and rearranging gives the formula. In practice, the prior P(A) is your starting belief, the likelihood P(B|A) measures how compatible the evidence is with that belief, and the denominator P(B) normalizes across all possible ways the evidence could occur.

Use Bayes' Theorem whenever you need to update a probability after observing new information, especially in classification, diagnostics, filtering, and decision-making under uncertainty. It is especially useful when the evidence is common or rare and you need to distinguish between "probability of evidence if a claim is true" and "probability the claim is true given the evidence."

• Confusing P(A|B) with P(B|A); they are usually not the same
• Forgetting to compute P(B) using all possible causes of the evidence
• Using Bayes' Theorem without checking whether the prior and likelihood are based on the same sample space