Forensics

 Let C be the donor of DNA to a stain. Suppose that a defendant S is accused of having left the stain. The stain, hence the DNA profile of C, has been determined on 20 autosomal loci, but the DNA profile of the defendant S has only been determined on 8 of those loci. It is found by the forensic laboratory that S and C have the same DNA profile on these 8 loci. The hypotheses that are under consideration are H1: C = S, and H2: C and S are unrelated. These are the only hypotheses with non-zero probability. The likelihood ratio in favour of H1, versus H2, equals 106 for the 8 compared loci. The population frequency of the profile of C restricted to the 12 loci that C is determined for, but S is not, is 10−10 . Suppose that the prior probabilities are π1 = P(H1) and π2 = P(H2) = 1 − π1. Compute the probability (in terms of π1, π2) that, if the genotype of S would be determined on the 12 remaining loci, we would find the same genotype as for C on all these loci.