Why does the rarity of the condition matter so much?

Because false positives are drawn from the healthy population, and when the condition is rare that population is almost everybody. A 1% error rate applied to 99,900 people produces far more positives than a 99% catch rate applied to 100.

Does this mean screening tests are useless?

No — it means a single positive is a reason to test again, not a conclusion. A second independent test applies the same arithmetic to a group that is now 9% affected rather than 0.1%, and the answer changes completely.

Where else does this show up?

Anywhere a rare thing is searched for with an imperfect filter: fraud detection, spam filtering, security screening, doping tests. The mathematics does not care what is being tested for.

Math

Bayes Theorem Calculator

Enter how common the condition is and how the test performs to see what a positive result really implies.

The formula

true positives = prevalence × sensitivity

false positives = (1 − prevalence) × false positive rate

answer = true positives ÷ (true positives + false positives)

Worked example

A condition one person in a thousand has, a test that catches 99% and wrongly flags 1%

Out of 100,000 people, 100 have it
99 of them test positive
Of the 99,900 who do not have it, 999 test positive anyway
99 ÷ (99 + 999) = 9.0% — a positive result is wrong nine times in ten

Where this goes wrong

Reading the test accuracy as your probability

A 99% accurate test does not mean a positive result is 99% likely to be right. When only 1 in 1,000 people has the condition, the 999 false positives among the healthy vastly outnumber the 99 true positives among the sick, and a positive means about a 9% chance. Rarity beats accuracy, and it beats it badly.

Questions

Why does the rarity of the condition matter so much?: Because false positives are drawn from the healthy population, and when the condition is rare that population is almost everybody. A 1% error rate applied to 99,900 people produces far more positives than a 99% catch rate applied to 100.
Does this mean screening tests are useless?: No — it means a single positive is a reason to test again, not a conclusion. A second independent test applies the same arithmetic to a group that is now 9% affected rather than 0.1%, and the answer changes completely.
Where else does this show up?: Anywhere a rare thing is searched for with an imperfect filter: fraud detection, spam filtering, security screening, doping tests. The mathematics does not care what is being tested for.