Chapter 9.2, Problem 66E

Gene mutation. Suppose that a gene in a chromosome is of type $A$ or type $B$. Assume that the probability that a gene of type $A$ will mutate to type $B$ in one generation is ${10}^{-4}$ and that a gene of type $B$ will mutate to type $A$ is ${10}^{-6}$.

(A) What is the transition matrix?

(B) After many generations, what is the probability that the gene will be of type $A$ ? Of type $B$ ? (Find the stationary matrix.)