As an aside, here's a funny variation of the Absent-Minded Driver problem that I just came up with. Nothing really new, but maybe it will make some features sharper.

You're invited to take part in an experiment, which will last for 10 days. Every day you're offered two envelopes, a red one and a blue one. One of the envelopes contains a thousand dollars, the other is empty. You pick one of the envelopes and receive the money. Also, at the beginning of each day you are given an amnesia pill that makes you forget which day it is, what happened before, and how much money you have so far. At the end of the experiment, you go home with the total money received.

The money is distributed between envelopes in this way: on the first day, the money has 60% chance of being in the red envelope. On each subsequent day, the money is in the envelope that you didn't pick on the first day.

Fun features of this problem:

1) It's a one-player game where you strictly prefer a randomized strategy to any deterministic one. This is similar to the AMD problem, and impossible if you're making decisions using Bayesian probability.

2) Your decision affects the contents of the sealed envelopes in front of you. For example, if you pick the red envelope deterministically, it will be empty >90% of the time. Same for the blue one.

3) Even after you decide to pick a certain envelope, the chance of finding money inside depends not just on your decision, but on the decision process that you used. If you used a coinflip, the chance of finding money is higher than if you chose deterministically.