Consider n independent uses of the binary symmetric channel with bit-flip probability p. The channel takes an n-bit vector as input X = (X0, X1, . . . , Xn) T and has n-bits vector an output Y = (Y1, Y2, . . . , Yn). Let X1, X2, . . . , Xn be the string (0, 0, ..., 0) with probability 1/2 and (1, 1, . . . , 1) with probability 1/2. This strategy is known as repetition coding, where the bit is simply repeated to reduce the probability of error. For this problem, we assume that n = 1 for part (a) and n = 3 for the remaining parts. (a) Assume n = 1. Find the optimal detection rule and the probability of error for n = 1.Your answer will depend on the bit flip probability p. (b) Assume n = 3. Note that there are 23 = 8 possibile outputs (Y1, Y2, Y3). For each of these outputs, determine the MAP decision. That is, given the output (y1, y2, y3) determine whether the input (0, 0, 0) or the input (1, 1, 1) has the greater aposteriori probability. Assume that p < 1/2. (c) Now observe your answer for part (b) and determine a simple decision rule for the channel based on the number of 1s in the output bit string. (d) Assume now that p > 1/2 and re-do parts (b) and (c).