the rare thing fools the test

Picture a test that's right ninety-nine percent of the time, pointed at something only one person in ten thousand has. The cleanest way to see the trap is to stop using percentages and just count people.

\[ \begin{aligned} \text{start with } 10{,}000 \text{ people} \quad & \\ \text{truly sick:}\quad 1 \;&\Rightarrow\; \text{caught} = 1 \\ \text{healthy:}\quad 9{,}999 \;&\Rightarrow\; \text{false alarms} = 1\% \times 9{,}999 \approx 100 \\ P(\text{sick}\mid +) &= \frac{1}{\,1 + 100\,} \approx 0.01 \end{aligned} \]

assume the test catches the sick person and wrongly flags 1 percent of healthy people, which is what "right 99 percent of the time" means here. P(sick given +) is the chance you're truly sick after a positive result. of about 101 people flagged, only 1 is actually sick, so a positive means roughly a 1 percent chance of being sick. how rare the condition is matters more than how accurate the test is.

When the thing is rare enough, almost everyone the test flags is a false alarm, because the large crowd of healthy people throws off more wrong positives than the tiny group of real cases produces right ones. This isn't just a medical thing. Any time you're checking for something rare, a positive result is weaker than it feels -- because there are so many more ways to get a false one than a true one.