1. When an engineer wants to send a binary signal (a 0 or 1 signal) through a communication channel, assume that there is a 10% chance that the wrong signal is recieved, that is, if the transmitted signal is a 1, then there is a 10% chance that the signal is a 0; while if the transmitted signal is a 0, then there is a 10% chance that the signal recieved is a 1.
In order to improve the signal transmitted reliablity, the signal to be transmitted will be sent five independent times to the reciever. The reciever will then interpret the signal to be a 1 if he/she recieves at least three 1's among the five transmissions, while he/she will interpret the signal to be a 0 if he/she recieves at least three 0's among the five transmissions.
(a) With the systme, what is the probability of getting the correct signal?
(b) Suppose you wanted to achieve at least 0.999999 reliability for your communication system. Denote by n the odd number of times that you will be transmitting your signal (in the proceding we took n=5), and assume that you will interpret the signal to be a 1 if you recieve at least (n+1)/2 1's among the n transmitted signals and 0 otherwise. What is the appropriate value of n?