Does the DNA test say guilty or not guilty?

Scenario

A young boy has been found murdered in England and crime scene investigators obtain a DNA sample on the boy’s body that belongs to someone else – another male person.

The police set up a wide network and collect the DNA of three-quarters of a million volunteers.

After a long period of massive testing they find no match to the DNA on the body.

Then, as the police scan their records, they notice a man who was living near where the body was found is not on their DNA volunteer list.

The man is eventually located in Portugal and his DNA tested. A positive match results.

Solely on the basis of the DNA testing, is the man more or less likely to be guilty?

Some assumptions

• The accuracy of the DNA test is 1 in one- million. That is, the probability the test returns a false positive is 0.000001
• The testing of volunteers is independent: that is, the results of the test of any one person is independent of the results of the test for any other person.

Calculation

Each DNA test is correct with probability 0.99999.

The probability that each of the 750,001 independent tests is correct is .

That is, it is more likely than not, that at least one of the DNA tests will prove to be a match, irrespective of whether any one of those tested committed the crime.

In other words the DNA evidence alone is insufficient to determine guilt to even likelihood, let alone to a high degree of probability.

Suspicions

Of course we are suspicious that the DNA of the man who was found, after he had not volunteered, matched that on the body. And we are suspicious because the man was found in Portugal after having left England.

But the totality of the DNA tests is not sufficient, of and by itself, to convict with a high degree of probability.

The paradox of testing less often

One of the medical examiners involved in the case reflected that the evidence might have worked better in favor of the prosecution  if they had tested fewer people.

For example, if the medical examiner had tested only 30,000 volunteers there would be a  chance that each of the DNA tests was correct.

At the other extreme – of only testing the suspect in Portugal – there would be a chance of being correct.

The moral, of course, is that even with very small probabilities of being wrong, when large numbers are tested it can quickly become more likely than not that a testing error – that is, a false positive –  is made.

Acknowledgments

I concocted this story from scenarios borrowed from Heather Dallas and David Mumford, and Tim Gowers (problem 45)