Bayes' Theorem & Base Rates

Imagine a screening test for a rare condition. The lab says the test is 90% accurate, and your result comes back positive. It is tempting to conclude there is a 90% chance you have the condition. That feeling is almost always wrong — and the gap between it and the truth is one of the most useful ideas in probability.

The trap is that 90% accurate describes \(P(\text{positive}\mid\text{have it})\): among people who have the condition, how often the test fires. But what you want to know is the reverse — \(P(\text{have it}\mid\text{positive})\): among people who test positive, how many truly have it. These are not the same number, and Bayes' theorem is the tool for turning one into the other.

Bayes' theorem stitches three quantities together:

• The base rate \(P(A)\) — often called the prior — how common the condition is before any test.
• The likelihood \(P(B\mid A)\) — how strongly the evidence B points to A (here, the test's true-positive rate).
• The overall chance of the evidence \(P(B)\) — how often a positive shows up at all, counting both true and false positives.

Put together, they give the flipped conditional you actually want:

Here is the heart of it. Suppose a condition affects just 1 in 100 people, and a test is 90% sensitive (catches 90% of real cases) and 92% specific (correctly clears 92% of healthy people). Those sound like strong numbers. Yet because 99 out of every 100 people are healthy, even a modest 8% false-positive rate on that huge healthy group produces a stack of false alarms — often more than the true positives from the tiny group who really have it.

Jumping straight from positive to I have it — while ignoring how rare the condition is — is the base-rate fallacy. The simulation below makes the pile of false positives impossible to miss. Drag prevalence down toward 1% and watch the red false-positive icons swamp the green true-positive ones, while the readout P(have it | positive) sinks far below the test's accuracy.

These problems get dramatically easier if you stop juggling percentages and instead picture a concrete crowd — say 1,000 people — and count heads. This trick is called using natural frequencies, and it turns Bayes' theorem into simple arithmetic you can do without the formula.