Consider the simple repetition code introduced on slide 3 of Lecture 12. In this code, three copies of each bit of data are transmitted sequentially: i. e., to transmit a 0, the signal 000 is sent; to transmit a 1, the signal 111 is sent. Suppose that these signals are sent over a particularly noisy channel in which each of the three copies of each transmitted bit has an independent 1% probability of being flipped during transmission. The receiver attempts to correct received codes by the principle of 'majority voting', so that, for example, a received signal ot 010 will be assumed to represent a 0 and a received signal of 110 will be assumed to represent a 1. Therefore, if zero or one of the bit copies is flipped during transmission, the receiver will find the correct value for the bit; if two or three bit copies are flipped, the receiver will obtain the wrong value.

Required:
a. Find the probability that any given transmitted bit of data is correctly decoded by the receiver. Show your working and briefly explain your method.
b. Now suppose that instead of transmitting three copies of each bit of data, a new coding scheme is used in which five copies of each bit of data are sent. The receiver is still programmed to use majority voting, so, e. g., a received sequence of 00011
would be decoded as a 0, and a sequence of 10101 would be decoded as a 1. What is now the probability of a transmitted bit of data being correctly decoded?