Past exam of the mathematics course of the University of Cambridge / 2006 / iii / Paper 45 / 3 / e / Solution

Let $D$ denote the specified disease pattern. With one introduced disease-allele copy, no new mutations or phenocopies, and fully penetrant recessive inheritance, every affected subject must carry two descendants of that one copy. Since every sibship has an affected child, all the upstream genetic carrier transmissions described above are forced. Conversely, unaffected siblings cannot carry two disease copies. Thus $D$ implies the inheritance event $E$ at the causal genetic locus, andThe rarity of the unconditional pattern in part (d) does not make its posterior probability small after observing the pattern that forces it. If the location of the one disease copy is initially uniform among the four founding copies, the likelihood of $D$ is the same for each, so its posterior location remains uniform; $P(E_q\mid D)=1/4$ for any named copy. If $q$ is already designated as the disease copy, its corresponding event has posterior probability 1.