Consider the following process. We have two coins, one of which is fair, and the other of which has heads on both sides. We give these two coins to our friend, who chooses one of them at random (each with probability 1/2). During the
rest of the process, she uses only the coin that she chose. She now proceeds to toss the coin many times, reporting the results. We consider this process to consist solely of what she reports to us.
Given that she reports a head on the nth toss, what is the probability that a head is thrown on the (n + 1)st toss?
Now assume that the process is in state "heads" on both the (n - 1)st and the nth toss. Find the probability that a head comes up on the (n + 1)st toss.