1.2. Consider the problem of sending a binary message, 0 or 1, through a signal channel consisting of several stages, where transmission through each stage is subject to a fixed probability of error

a. Suppose that X, = 0 is the signal that is sent and let X,, be the signal that is received at the nth stage. Assume that (X) is a Markov chain with transition probabilities Poo P 1a and Po = Pio =

a, where 0 < a < 1.

(a) Determine Pr{X = 0, X = 0, X = 0}, the probability that no error occurs up to stage n = 2.

(b) Determine the probability that a correct signal is received at stage 2. Hint: This is Pr{X = 0, X = 0, X = 0} + Pr{X = 0, X = 1, X = 0}.