A one-in-a-thousand DNA match

Question

A DNA test always matches the true source of a sample, and it also matches a random innocent person with probability $\tfrac{1}{1000}$ . Police run the test against a database of 20,001 people that happens to include the one true source (so 20,000 are innocent). Everyone in the database is tested, and several people match. You pick one of the matched people at random.

What is the probability that this person is the true source?

Accepted Answer

Count how many matches you expect, of each kind. - The true source: matches for certain, 1 person. - Innocent people: 20,000 of them, each matching with probability 	frac{1}{1000}, giving about 20{,}000 	imes 	frac{1}{1000} = 20 expected false matches. So roughly 1 + 20 = 21 people match, and only 1 of them is the source. Picking a matched person at random, P(	ext{source} \mid 	ext{match}) \approx \frac{1}{21} \approx 4.8\%. The "one in a thousand" match rate sounds damning for any single innocent person, but searched against 20,000 innocents it produces about 20 coincidental matches, sw…

A one-in-a-thousand DNA match

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