21. A DNA nucleotide has any of 4 values. A standard model for a mutational change of the nucleotide at a specific location is a Markov chain model that supposes that in going from period to period the nucleotide does not change with probability 1 − 3α, and if it does change then it is equally likely to change to any of the other 3 values, for some 0 <α< 1 3 .

(a) Show that Pn 1,1 = 1 4 + 3 4 (1 − 4α)n.

(b) What is the long run proportion of time the chain is in each state?