So let's say you and I sit down, and do the following: 1) I roll a 3-sided die (or a 6 sided, wrapping) and keep it covered. 2) You pick a number, 1-3

So let's say you and I sit down, and do the following: 1) I roll a 3-sided die (or a 6 sided, wrapping) and keep it covered. 2) You pick a number, 1-3 3) I flip a coin. If it's heads, I pick the lower available number; if it's tails I pick the higher. 4) I peak at the die. If it's my number, we reveal and start over. 5) I (always, at this point) offer you a wager: if the die shows your number (so you would have lost if you switch), you pay me $7; if the die doesn't show your number, I pay you $5. Assuming you believe the die and coin are fair, etc, would you agree to play that game 1000 times? In those games, is there a reason you would turn down the wager? Is it the same game if I flip the coin secretly, peek at the die, announce my number (picked algorithmically in the obvious way), and then pay out according to whether switching would win (as above)? Because (assuming I've explained these games as I intend... it's getting late) I would play the former with you not the latter (but I would play the latter if we switched the payments around - I chose 5 and 7 because 7/12 is halfway between 1/2 and 2/3).