In many cases, you can model a binary communication system as follows. A 0 or 1 is
chosen as the message and the transmitter sends a signal over a medium to indicate
that choice. The receiver receives a corrupted version of that signal and makes the
best decision it can about whether a 0 or 1 was intended. Suppose that the message
is a 1 with probability p. Also suppose that no matter what the transmitted message
was, the receiver makes a random decision error about the transmitted message with
a probability . Let A be the event that the intended message was a 1, and B be the
event that the receiver decision was a 1.

(a) There are four possible outcomes for this experiment. List them along with their
associated probabilities.

(b) Find a value for  for which the intended message and received message are
independent. Comment on the value of this as a practical communication system.