Bayes Theorem
Reverses conditional probabilities using prior knowledge.
Formula
Variables
Example
Core of spam filters and medical testing
How it works
Bayes’ theorem lets you update the probability of a hypothesis A after seeing evidence B. It comes directly from the definition of conditional probability: the chance of A and B together can be written as P(A|B)P(B) or P(B|A)P(A), so rearranging gives the formula. The key idea is that evidence is not interpreted in isolation; it is weighed against how likely that evidence is under different possible causes.
When to use it
Use this when you know how likely some evidence is if a hypothesis is true, but you want the reverse: how likely the hypothesis is given the evidence. It is especially useful in diagnostic testing, classification, fault detection, and any problem with competing explanations.
Worked example, step by step
Suppose 2% of a factory’s widgets are defective. We test a widget using a detector that is 95% accurate for defective items, so P(test + | defective) = 0.95. The false positive rate is 4%, so P(test + | not defective) = 0.04. We want P(defective | test +). Step 1: Find the overall probability of a positive test: P(test +) = P(test + | defective)P(defective) + P(test + | not defective)P(not defective) P(test +) = (0.95)(0.02) + (0.04)(0.98) P(test +) = 0.019 + 0.0392 = 0.0582 Step 2: Apply Bayes’ theorem: P(defective | test +) = [P(test + | defective)P(defective)] / P(test +) P(defective | test +) = (0.95 × 0.02) / 0.0582 P(defective | test +) = 0.019 / 0.0582 ≈ 0.3265 Final answer: P(defective | test +) ≈ 0.327, or about 32.7%.
Common mistakes to avoid
- Confusing P(A|B) with P(B|A): the order matters and the two are usually not equal
- Forgetting to calculate P(B) as the total probability across all ways B can happen
- Ignoring the prior probability of A, which can strongly change the result even with strong evidence
Did You Know?
Bayes theorem underpins much of modern statistics and machine learning.
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