Invitation to a Beheading

And also to confusing myself 100% of the time. Anyway, questions in real life: is Monty providing me with relevant information at all (which he is not if he is just arbitrarily eliminating a door that actually has a 1 in 3 chance of having the car behind it), and if so, how confident should I be in the information he's providing? Suppose Monty says, look, of those other two doors you didn't pick, I can tell you I'm 80% confident there's no car behind door #3 ... and suppose I'm 90% confident that Monty is sincere, and I'm 90% confident in Monty's sincerely-expressed judgments ... so I'm 90% confident that I'm 90% confident that there's an 80% chance there's no car behind door #3 ... which entails I ought to be 64.8% confident there's no car behind door #3 ... which, sun of a gun, means I ought to switch to the door Monty told me he's 80% sure there's no car behind, because it has a 35.2% chance of winning, my original door still has its 33.3...% chance of winning, and door #2 now holds down the remaining 31.5%. The correctness of which I am let's say 50.5% confident in. (On the other hand, supposing the same stuff about my confidence in Monty but supposing that Monty tells me that he's 80% confident that if one of doors #2 and #3 is the winner then it's #3, then I ought to switch to door #3, which I ought to be about 43.2% confident is actually the winner. Whew.)

Really it's amazing we ever do better than chance on anything. Or that we don't manage to outsmart ourselves into doing worse than chance on everything.