Past exam of the mathematics course of the University of Cambridge / 2002 / iii / Paper 57 / 6 / a

Take “cousins” to mean first cousins with unrelated, outbred founders and unrelated spouses in the intervening generation. Interpret a homozygous chromosome as two copies identical by descent, and first use an idealization of independent, nonrecombining chromosome blocks. Ordinary sequence homozygosity is a different quantity: unrelated copies can have the same allele, so its probability also requires allele frequencies. Similarly, identity of an entire recombining chromosome requires a genetic map, not just a pedigree.

At a single autosomal locus, the probability of autozygosity in their child is the parents' kinship coefficient. Each of the two common grandparents contributes a path of length two from each parent, with probability $(1/2{)}^{2+2+1}$. Thus the first-cousin inbreeding coefficient isIf there are $L$ independent autosomal blocks, the probability that none is autozygous is $(1-F{)}^L$. Hence the usual 23-block idealization givesThe exponent describes the stated independent-block approximation; it is not a claim that all 23 human chromosome pairs inherit like autosomes.

Humans have 22 autosomal pairs. A son has one X and one Y and cannot have two homologous X copies. A daughter can have an autozygous X, with probability $F_X$ determined by the sex-linked cousin kinship calculation in part (b). Thus a sex-aware version iswhere $p_D$ is the probability of a daughter, or is one or zero for a specified daughter's or son's sex. The independent assortment approximation applies between the autosomes and the X. Opposite-sex parentage alone does not determine $F_X$: the parents may be cousins through their mothers, their fathers or one of each. Without these pedigree and chromosome-block assumptions, the wording does not specify a unique probability of an entire homozygous human chromosome.