Question

A person commits a crime in a city with a population of 5 million. A suspect is identified and arrested. Moreover, the suspect’s DNA matches a sample found at the crime scene. The odds of an erroneous DNA match are one in 1 million. At trial, the prosecutor uses this one-in-a-million statistic to argue that the suspect is guilty. However, the suspect’s defense attorney is well-trained in statistics, and is able to show that there is in fact an 83% chance that an innocent person’s DNA would match the sample found at the crime scene. How did he arrive at this conclusion?

Answer 1

Let M and ~M be the event that DNA match occur and DNA match not occur respectively.

Let In and ~In be the event suspect is innocent and not innocent respectively.

As, there is only one person that commited a crime in population of 5 million.

, the probability of a person being not innocent is P(~In) = 1 / 5 million

P(In) = 1 - (1 / 5 million) 1

The odds of an erroneous DNA match are one in 1 million. So, given an innocent person, the probability of DNA match is, P(M | In) = 1/ 1 million

The odds of an correct DNA match are 1 - (1/ 1 million). So, given an not innocent person, the probability of DNA match is, P(M | ~In) = 1 - (1/ 1 million) 1

By law of total probability, the probability of DNA match is,

P(M) = P(In) * P(M | In) + P(~In) * P(M | ~In)

= 1 * (1/ 1 million) + (1 / 5 million) * 1

= 6 / (5 million)

Given DNA matched, the probability that the person is innocent is,

P(In | M) = P(M | In) * P(ln) / P(M)

= [(1/ 1 million)] * 1 / [ 6 / (5 million)]

= (5 million) / (6 * 1 million)

= 5/6

= 0.8333

= 83.33%

Thus, there is 83% chance that an innocent person’s DNA would match the sample found at the crime scene.