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 received, that is, if the transmitted signal is a 1, then there is a 10% chance that the signal received is a 0; while if the transmitted signal is 0, then there is a 10% chance that the signal received is a 1.
In order to improve the signal transmission reliability, the signal to be transmitted will be sent five independent times to the receiver. The receiver will then interpret the signal to be a 1 if he/she receives at least three 1's among the five transmissions, while he/she will interpret the signal to be 0 if he/she receives at least three 0's among the five transmissions.
a. With this system, what is the probability of getting the correct signal? [This probability is called the reliability of the communication or transmission system.] Is the probability high enough?
b. Suppose you wanted to achieve at least a 0.999999 reliability for your communication system. Denote by n the odd number of times that you will be transmitting your signal (in the preceding we took n=5), and assume that you will interpret the signal to be a 1 if you receive at least (n+1)/2 1's among the n transmitted signals, and 0 otherwise. What is the appropriate value of n?