Imagine a crime scene where the investigators were able to collect bloodstain. The sample was old, the DNA degraded, and the analysts estimated a relative frequency of 1 in 1000 in the population. Police found a suspect and got a DNA match. What is the chance that the suspect is guilty?
The prosecutor argues that since the relative frequency of the DNA match is 1 in 1000, the chance for the person to be innocent is 1 in 1000 and deserves maximum punishment. Well, the prosecutor made a wrong argument here. Imagine the city has 100,000 people in it. The test results suggest that there are about 100 people whose DNA can match the sample. So, the suspect is one of 100, and the chance of innocence only based on the DNA test is 99%.
Not Convinced? Let us use Bayes’ theorem.
P(INN|DNA) = P(DNA|INN)*P(INN)/ [P(DNA|INN)P(INN) + P(DNA|CRI)*P(CRI)]
P(INN|DNA) – the chance that the suspect is innocent given the DNA matches
P(DNA|INN) – chance of DNA match if the suspect is innocent = 1/1000
P(CRI) – prior probability that the suspect did the crime = 1 /100,000 (like any other citizen)
P(INN) – prior probability that the suspect is innocent = (1 – 1 /100,000)
P(DNA|CRI) – chance of DNA match given the suspect did the crime = 1 (100%)
P(INN|DNA) = (1/1000)* (1 – 1 /100,000) / ((1/1000)* (1 – 1 /100,000) + 1*(1/100000)) = 0.99
Does this mean that the suspect is innocent? Not either. The results only mean that the investigators must collect more evidence to file charges against the suspect.
