A DNA nucleotide has any of four 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 three 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?