# DNA Test: Probability question — matter of interpretation?

A crime has been committed in a town of 100,000 inhabitants. The police are looking for a single perpetrator, believed to live in the town. DNA evidence is found on the crime science. Kevin's DNA matches the DNA recovered from the crime scene. According to DNA experts, the probability that a random person's DNA matches the crime scene DNA is 1 in 10,000. Before the DNA evidence, Kevin was no more likely to be the guilty person than any other person in town. What is the probability that Kevin is guilty after the DNA evidence appeared?

Kevin is no more likely to be the perpetrator than any one else in the town. I AM assuming that they checked the DNA evidence against every town inhabitant (otherwise, they thought Kevin was more likely than any other town inhabitant to be the perpetrator and only tested it against his, and maybe against that of a few others). So, assuming 1 in 10,000 have the 100,000 came back positive, Kevin is one among ten inhabitants whose DNA matched with that found on the crime scene. Therefore, there is a $\frac{1}{10}$ probability that Kevin is the criminal.

Solution given:

• Please try to make the titles of your questions more informative. For example, Why does $a<b$ imply $a+c<b+c$? is much more useful for other users than A question about inequality. From How can I ask a good question?: Make your title as descriptive as possible. In many cases one can actually phrase the title as the question, at least in such a way so as to be comprehensible to an expert reader. You can find more tips for choosing a good title here. – Shaun Sep 12 '18 at 23:11
• If we tested everyone, we expect about 10 false positives (more precisely $\frac {99,999}{10,000}$) and one true positive for 11 total positive results. Of which, Kevin is one. – Doug M Sep 12 '18 at 23:14
• Thanks Shaun and Doug! – Rafael Vergnaud Sep 12 '18 at 23:36
• Except, it doesn't say anything about false positives. It just says a random test will have a chance of 1/10000. – Rafael Vergnaud Sep 12 '18 at 23:38
• " Kevin is no more likely to be the perpetrator than any one else in the town" and " Therefore, there is a 1/10 probability that Kevin is the criminal". You realize those are utterly contradictory statements, don't you? – fleablood Sep 12 '18 at 23:50

There is some ambiguity about whether the "1 in 10,000" fraction refers to any random person or to any random innocent person. The given solution is assuming the latter (notice it says $P(A\mid G^c)=\frac1{10000}$). If you also assume the latter, then your argument will agree with the solution: the guilty person will test positive, and so will $\frac1{10,000}$ of the remaining 99,999 townsfolk, meaning 10.9999 people in total will come back positive. Kevin is one person in that total.

• Well the question said "random person" so is that assumption in the solution wrong? Also, what would $P(A|G^c)$ be if the former was assumed? – Tomás Palamás Feb 28 at 3:41
• Would $P(A|G^c)=\frac{9}{100000}$? – Tomás Palamás Feb 28 at 3:59

One way to get a better grasp on this problem is to think of the four possibilities:

1. Kevin is guilty and the DNA test is not a match
2. Kevin is guilty and the DNA test is a match
3. Kevin is not guilty and the DNA test is not a match
4. Kevin is not guilty and the DNA test is a match.

Once the DNA test comes back as a match, we're restricting ourselves to situations 2 and 4, which leads to the calculation as given by the textbook.

Even if the police test everyone, you would still expect about 10 people from the 99999 who aren't Kevin to show up as positive, leading Kevin to be one of (about) 11 people who test positive.

• Why can't it be interpreted that Kevin in simply one of those 10 people of the 100000 that come back positive? – Rafael Vergnaud Sep 12 '18 at 23:40
• All we are told is that Kevin came back positive. But he was randomly tested. And so would everyone else have been. Meaning that 10 people came back positive. – Rafael Vergnaud Sep 12 '18 at 23:40
• No, this is not true. All we know is that Kevin was tested: we don't know that anyone else was tested. And even if everyone else was tested, we would still expect to find about 10 (really, 99999/10000) people who also tested positive from the people who aren't Kevin. – KReiser Sep 12 '18 at 23:46
• Ok. I think this shed's light on my confusion. Why was Kevin tested as if he were not equally likely among all inhabitants to be the criminal? From the account, is it not fair to assume that if Kevin is tested, EVERYONE is tested? – Rafael Vergnaud Sep 12 '18 at 23:51
• Don't forget the killer will test positive as well. So there will be 9.9999 innocent peple who test positive and 1 person who tested positive. That's 11 total. – fleablood Sep 12 '18 at 23:56