The Prosecutor\'s fallacy. The probability that there is a DNA match given that

Question

The Prosecutor's fallacy. The probability that there is a DNA match given that a person is innocent is estimated as 1/100,0007. Assume that the probability that there is a match given that a person is guilty is equal to 1. Suppose that the defendant in a trial lives in a city where there are 10,000 people that could have committed the crime, and that there is a DNA match to the defendant. Calculate the probability that the defendant is indeed guilty, given no other evidence except the DNA match, i.e., P(guilty | DNA match). If the city would be New York, what is the probability that the defendant is indeed guilty, given no other evidence except the DNA match?

Explanation / Answer

a) P(guilty | DNA match) = P(guilty and DNA match) / P(DNA match)

= [1x(1/10,000)]/[(1x(1/10,000)) + (9999/10000)x(1/100,000)] = 0.9091

b) Take the population of new york as 8,000,000

P(guilty | DNA match) = [1x1/8,000,000]/[(1x1/8,000,000) + (7,999,999/8,000,000)x(1/100,000)]

= 0.1111

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