Consider a disease trait partially determined by an autosomal locus with two alleles 1 and 2 having frequencies p1 and p2. Let φk/l be the probability that a person with genotype k/l manifests the disease.

For the sake of simplicity, assume that people mate at random and that the disease states of two relatives i and j are independent given their genotypes at the disease locus. Now let Xi and Xj be indicator random variables that assume the value 1 when i or j is affected, respectively. Show that Pr(Xj = 1 | Xi = 1) = gi
gj
Sr Pr(Xj = 1 | gj ) Pr(gj | Sr, gi)
× Pr(Sr | gi) Pr(gi | Xi = 1), (6.9)
where gi and gj are the possible genotypes of i and j and Sr is a condensed identity state. This gives an alternative to computing risks by multiplying the relative risk ratio λR by the prevalence K. Explicitly evaluate the risk (6.9) for identical twins and parent–offspring pairs.