An urn contains two 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 color of the ball (from red to black or from black to red). to model this situation as a stochastic process, we define time t to be the time after the coin has been flipped for the tth time and the chosen ball has been painted. the state at any time may be described by the vector [u r b], where u is the number of un-painted balls in the urn, r is the number of red balls in the urn, and b is the number of black balls in the urn. we are given that x0 = [2 0 0]. after the first coin toss, one ball will have been painted either red or black, and the state will be either [1 1 0] or [1 0 1]. hence, we can be sure that x1 = [1 1 0] or x1 = [1 0 1]. clearly, there must be some sort of relation between the xt’s. for example, if xt = [0 2 0], we can be sure that xt+1 will be [0 1 1]. determine the following n-step transition probabilities:

a) after two balls are painted, what is the probability that the state is [0 2 0]?

b) after three balls are painted, what is the probability that the state is [0 1 1]?