An urn contains three unpainted balls at present. We choose a ball at random and flip 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?