  Many examples of exactly solvable birth and death processes, a typical
stationary Markov chain, are presented together with the explicit expressions
of the transition probabilities. They are derived by similarity transforming
exactly solvable `matrix' quantum mechanics, which is recently proposed by
Odake and the author. The ($q$-)Askey-scheme of hypergeometric orthogonal
polynomials of a discrete variable and their dual polynomials play a central
role. The most generic solvable birth/death rates are rational functions of
$q^x$ ($x$ being the population) corresponding to the $q$-Racah polynomial.
