[Part of this exercise was lifted from [37].] A = B = {a,b,c}, a is always transmitted correctly, and the channel behaves symmetrically with respect to b and c. Each has probability p of being transmitted correctly, and probability 1- p of being received as the other character (c or b). (Thus, if a is received, it is certain that a was sent.)
(a) Find the capacity of this channel, and the optimal input frequencies, as functions of p.
(b) Suppose that c is omitted from the input alphabet (but not the output alphabet). Find the capacity of the channel and the optimal input frequencies in this new situation.
(c) Are there any values of p for which the capacity found in (b) is greater than that in (a)? What about the case p = 1/2?