In Class: Bayes’ Rule and DNA Testing

Introduction

Forensic DNA evidence has become an important tool in criminal trials, used frequently by both prosecution and defense. Although DNA is unique to each individual, damage to samples as well as human error mean that DNA testing can still generate false matches with individuals entirely unrelated to the situation at hand.

Suppose that a DNA test has a 0.1% false positive rate. This means that, in a population of 1,000 people who did not contribute that DNA sample, testing will still erroneously match one of them (on average). We’ll assume that DNA testing has a true positive rate of 100%: if a sample was contributed by an individual, DNA testing will always name that individual as one of the possible people who could have contributed that sample.

This number, like every other element of this problem, is intended to be somewhat realistic but is fiction.

During late spring, an unexpected wildfire breaks out in central Vermont. After several days, investigators visit the site of the fire and find a DNA sample, dated to around the time the wildfire started. Testing matches the sample to a person who was known to be visiting the central Vermont region during the time the fires started. The case is brought to court, with the prosecution claiming that the visitor started the fire.

Part A

Let \(A\) be the event that the sample matches the suspect under DNA testing. Let \(B\) be the event that DNA sample truly belongs to the suspect.

Compute the probabilities \(p(A|B)\) and \(p(A|\bar{B})\) given the information about the reliability of DNA testing supplied above.

Part B

The prosecution argues that, since the false-positive rate of DNA testing is 1/1,000 = 0.1%, the probability \(p(B|A)\) that the DNA sample truly belongs to the suspect is 99.9%.

What is wrong with the prosecution’s argument?

Part C

A statistical expert for the defense testifies that 1,000 people visited the site of the wildfire during the time-window in question, any of whom could have left a DNA sample.

Use Bayes’ rule to compute an estimate of the probability that the suspect did indeed contribute the DNA sample, given that DNA testing produced a match to the suspect.