Suppose that a communications network transmits binary digits, 0 or 1, where each digit is transmitted 10 times in succession. During each transmission, the probability is 0.99 that the digit entered will be transmitted accurately. In other words, the probability is 0.01 that the digit being transmitted will be recorded with the opposite value at the end of the transmission. For each transmission after the first one, the digit entered for transmission is the one that was recorded at the end of the preceding transmission. If X0 denotes the binary digit entering the system, X1 the binary digit recorded after the first transmission, X2 the binary digit recorded after the second transmission, . . . , then {Xn} is a Markov chain.

(a) Construct the (one-step) transition matrix.

C

(b) Use your OR Courseware to find the 10-step transition matrix P(10). Use this result to identify the probability that a digit entering the network will be recorded accurately after the last transmission.

C

(c) Suppose that the network is redesigned to improve the probability that a single transmission will be accurate from 0.99 to 0.999. Repeat part

(b) to find the new probability that a digit entering the network will be recorded accurately after the last transmission.