An urn contains three unpainted balls at present. We choose a ball at

random and ip a coin. If the chosen ball is unpainted and the coin comes

up heads, we paint the chosen unpainted ball red; if the chosen ball is

unpainted and the coin comes up tails, we paint the chosen unpainted

ball black. If the ball has already been painted, then (whether heads or

tails has been tossed) we change the colour of the ball (from red to black,

or from black to red).

(a) Define the state variable(s), and determine the transition probability matrix.

(b) What is the probability after two steps to have 1 ball unpainted, 1 red ball and 1 black ball?