What Is Bayes' Theorem and How Do You Apply It?

Bayes' theorem calculates a conditional probability in the reverse direction: it finds the probability of a cause or group after observing new evidence. In IB Mathematics: Analysis and Approaches HL, it is part of AHL 4.13 Bayes' theorem and covers a maximum of three events.

Bayes' theorem is

$$P(A\mid B)=\frac{P(B\mid A)P(A)}{P(B)}.$$

This works because Bayes' theorem reweights the prior probability using how strongly the observed evidence supports each possible cause. The denominator ensures that the updated probabilities are measured against every route that could have produced the evidence.

The denominator often comes from the total probability of observing $B$. If $A$ and (A') partition the sample space, then

$$P(B)=P(B\mid A)P(A)+P(B\mid A')P(A').$$

Suppose $2\%$ of a population has a condition. A test gives a positive result for $95\%$ of people with the condition and falsely gives a positive result for $4\%$ of people without it. Let $C$ represent having the condition and $+$ represent a positive result.

First calculate the probability of a positive result:

$$P(+)=P(+\mid C)P(C)+P(+\mid C')P(C')$$ $$=(0.95)(0.02)+(0.04)(0.98)=0.0582.$$

Then apply Bayes' theorem:

$$P(C\mid +)=\frac{(0.95)(0.02)}{0.0582}\approx 0.326.$$

A frequency interpretation gives the same result. Out of $10000$ people, about $200$ have the condition, and $190$ test positive. Of the remaining $9800$, about $392$ test positive falsely. Thus, $190$ of $582$ positive results are true positives.

A common misconception is that $P(A\mid B)=P(B\mid A)$. These probabilities are generally unequal. Confusing them ignores the base rate, $P(A)$, which can make the posterior much smaller than the test accuracy.

In an IB AA HL response, define the events, show the total-probability denominator, substitute into Bayes' theorem, and give the final probability exactly or to three significant figures. A tree diagram may also provide an efficient route.