[math] \newcommand{\NA}{{\rm NA}} \newcommand{\mat}[1]{{\bf#1}} \newcommand{\exref}[1]{\ref{##1}} \newcommand{\secstoprocess}{\all} \newcommand{\NA}{{\rm NA}} \newcommand{\mathds}{\mathbb}[/math]

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 [math]n[/math]th toss, what is the probability that a head is thrown on the [math](n+1)[/math]st toss?
• Consider this process as having two states, heads and tails. By computing the other three transition probabilities analogous to the one in part (a), write down a “transition matrix” for this process.
• Now assume that the process is in state “heads” on both the [math](n-1)[/math]st and the [math]n[/math]th toss. Find the probability that a head comes up on the [math](n+1)[/math]st toss.
• Is this process a Markov chain?