The king who wants to test everyone

Question

A kingdom has one million people, and one in every 500 has a certain disease. A test for the disease has a false-negative rate of 1 per cent (a sick person tests negative 1 time in 100) and a false-positive rate of 2 per cent (a healthy person tests positive 2 times in 100). The king plans to test everyone and quarantine every positive. 1. How many people will test positive? 2. Of those, what fraction actually has the disease?

Accepted Answer

About 21,940 people test positive, and only about 1 in 11 of them is actually sick. Split the population. One in 500 of a million is 2,000 sick people and 998,000 healthy people. Apply the test. - Sick people: 99 per cent test positive: 0.99 × 2,000 = 1,980 true positives (20 are missed). - Healthy people: 2 per cent test positive: 0.02 × 998,000 = 19,960 false positives. Totals. Positives: 1,980 + 19,960 = 21,940. Of these, only 1,980 are sick, a fraction of 1,980 / 21,940, about 9 per cent, roughly 1 in 11. Why the test looks so bad despite being 98 to 99 per cent accurate. The disease is r…

The king who wants to test everyone

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