The Wright-Fisher Model of Genetic Drift

Problem

Consider flipping a weighted coin that gives "heads" with some fixed probability p (i.e., p is not necessarily equal to 1/2).

We generalize the notion of binomial random variable from “Independent Segregation of Chromosomes” to quantify the sum of the weighted coin flips. Such a random variable X takes a value of k if a sequence of n independent "weighted coin flips" yields k "heads" and n−k "tails." We write that X∈Bin(n,p).

To quantify the Wright-Fisher Model of genetic drift, consider a population of N diploid individuals, whose 2N chromosomes possess m copies of the dominant allele. As in “Counting Disease Carriers”, set p=m2N. Next, recall that the next generation must contain exactly N individuals. These individuals' 2N alleles are selected independently: a dominant allele is chosen with probability p, and a recessive allele is chosen with probability 1−p.

Given: Positive integers N (N≤7), m (m≤2N), g (g≤6) and k (k≤2N).

Return: The probability that in a population of N diploid individuals initially possessing m copies of a dominant allele, we will observe after g generations at least k copies of a recessive allele. Assume the Wright-Fisher model.