A gene in a living cell of a primitive organism consists of r units, m of which are mutants and the rest are normal. Prior to each division of the cell, the gene duplicates. The genes in each of the two daughter cells are assumed to consist of r units chosen at random from the 2m mutant units and the 2(r m) normal units of the two duplicate genes. Let x(u, n) be the number of mutant units in the n-th generation daughter cell in a particular line of descent.

(a) Show that x(u, n) is a Markov chain.

(b) Is the chain homogeneous? Is it stationary? Classify the states of this chain.

(c) For r = 3, what is the probability of first return of x(u, n) in k transitions to a state of m = 2 mutant units? Describe how you would do this computation for arbitrary k. Optionally, e.g., if you want to play with MATLAB, come up with the solution for arbitrary k.