If there is a 7.5% chance that two people share one spike (locus) of DNA, what is the chance two people share nine loci? Well, let it be (7.5/100)9 = 7.5 x 1011 or 1 in 13 billion! So a decent case for DNA match as forensic evidence!
Now the twist, an Arizona laboratory reported about 100 matches with nine loci of DNAs in a database of just over 60,000 samples. How is that possible? The first (1 in 13 billion) was an estimate, and this is data. So the estimation must be wrong by a zillion miles, right?
If you recall the birthday problem, you may realise this can’t be dismissed without further enquiry. Let’s start
Suppose there are 60,000 samples. What is the number of distinct pairs that can form from 60000? It is 60000C2 = 60000 x 59999 / 2 = 1,799,970,000. For each pair, how many ways to match 9 out of 13 loci? It is 13C9 = 13!/(4! x 9!) = 715. So the total number of 9 loci match = 1,799,970,000 x 715 = 1.286979 x 1012.
If the chance of 9 local matches of one pair is 1 in 13 billion, then the number of matches possible in 1.286979e+12 pairs is 1.286979 x 1012/13 x 109 = 99.
