Fantastic answer. I went through and built the tree and it just shows everything so neatly. There's just a small mistake, when you say: 4. then have the c

Fantastic answer. I went through and built the tree and it just shows everything so neatly. There's just a small mistake, when you say: 4. then have the contestant either switch or not switch and cut all your probabilities in half again. You don't halve probabilities again because the strategy we are evaluating is _to switch_. So we always switch. It is deterministic. If we switch deterministically we end up with 6 cases (leaves in the tree) with probability 1/9 where we win. 6/9 = 2/3! If we switch randomly then we do split probabilities again, and end up with 6 cases with probability 1/18. 6/18 = 1/3.