This exercise will focus on understanding the role of probability in communication channels. Alice wants to send Bob a message M composed of N:100 bits. This message is a length N sequence of 0's and 1's. The challenge is that the communication channel between Alice and Bob can introduce errors by ipping 0 to 1 or vice versa. We are given that the channel ip probability is p201. In other words, if Alice transmits a: 6 {0,1}, Bob receives a: \"nth .pmbablhtlf 1p 1 as w1th probability 19 Step 1 encoding: To safely transfer the message, Alice applies encoding by repeating the bits R times and transmits the encoded message Mme. For instance, for N = 3 and R = 3, an example message is M = 0 l 0 > encoding > Mam = 000 111 000 Step 2 transmission: After transmission, Bob receives a message Mmc which is a corrupted version of Mam due to channel ips. For instance, we might have Mam = 000 111 000 > channel > M\"c = 001 101 110 Step 3 decoding: Now, Bob needs to decode Mm: to nd M. His goal is ensuring M = M Let us use \"majority rule\" for decoding: Bob assigns each Rbit chunk to 0 or 1 by counting which one is more. For instance, we assign 001 > 0 whereas 111 > 1. For the example above, this means Mme = 001 101 110 > decoding > M = 0 1 1 Unfortunately, for this example M at M at the end of the encoding, transmission, and decoding process. This HW aims to understand the probability of successful communication as a function of R, N and 19. Our goal is understanding what makes the communication process safe. You are expected to answer following questions by doing numerical simulations. You should turn in your report explaining how you arrived at your answers and also turn in your code. 1. (3 pts) Write a code for generating the procedure described above. Your code should allow for any p, N, and R values. You can assume R is an odd integer. Your code should be able to carry out one communication experiment and output whether all bits of Alice was correctly transmitted to Bob. 2. (1 pts) Set N = 10,13 = 9,;0 = 0.1 and run your experiment for 1000 times. Record the resulting probability of successful decoding pauccess = P(M = M). 3. (2 pts) Now try N = 30, 100,300, 1000. Record the resulting probability of successful decoding psuccegg. Plot psmcess as a fimction of N and comment on how it changes. 4. (1 pts) Set p = 0.2 and N = 100. Suppose we wish to ensure psmcess 2 0.9. What is the minimum R we need to choose? You should nd your R choice by running sufciently many communication experiments. 5. (2 pts) How about for ensuring psaccess 2 0.99 and psuccess 2 0.999? 6. (1 pts) How does pmmss change as a function of N, R, p? You should explain your answer (why it is increasing or decreasing)