The probability that related individuals share some section of genome identical by descent

A formal mathematical framework is presented for the study of linkage in man and the concept of chromosome pedigree is defined for both autosomes and X chromosomes. It is shown that, assuming no interference, all the crossover processes in the pedigree may be viewed jointly as a continuous-time Markov random walk on the vertices of a hypercube, the time parameter being map distance along the chromosome. The event that two individuals have a segment of chromosome in common, thus proving them to be related, corresponds to the random walk hitting a particular set of vertices. The probability of this happening is calculated for various types of relationship, making use of the symmetry of the situation to partition the vertices into a very much smaller number of orbits and render the computation manageable. The probability that an individual with n children passes on all his or her genes to them is also calculated in this way.