1. ( Communication through a Noisy Channel) A source transmits a message (a string of symbols) through a. noisy communication channel. Each symbol is O or 1 with probability p and 1 19, respectively. And each symbol is received incorrectly with probability 60 and 61, respectively (see Figure 1). Errors in different symbol transmissions are independent. (a) What is the probability that the k-th symbol is received correctly? (b) What is the probability that the string of symbols 1011 is received correctly? (c) To improve reliability, each symbol is transmitted three times and the received string is decoded by majority rule. In other words, a 0 (or 1) is transmitted as 000 (or 111) and it is decoded at the receiver if and only if the received threesymbol string contains at least two US (or two 1s). What is the probability that a 0 is correctly decoded? ((1) Suppose the three-symbol scheme is used. What is the probability that a symbol was 0 given that the received string is 101