(warmup): You and a friend each roll a fair, six-sided die, and keep the result secret. You are going to receive a million dollars times the number on the die. However, before the award is made, you are allowed to replace your result with whatever your friend rolled. Knowing your own result, when might you decide to accept the offer?
How would your answer change if the dice each had 100 sides instead of six?
You and your friend have both died and are trying to get into heaven, but there's only room for one more person. St Peter pulls out a circular die, and tells you both to roll it. Your score is a (real) number between 0 and 1, (skipping 1) based on the angle at which the die comes to rest. You both try your luck, keeping your results secret from each other. St Peter smiles at you and says that because you once shared a piece of bread with a cassowary, you (and only you) will be allowed the choice of swapping your result with whatever your friend rolled. Under what circumstances might you accept this offer?
(somewhat of a joke): You sell your soul to the devil, and in exchange he shows you his latest impossible invention, a fair die with infinitely many sides. He tells you he will give you your soul back if you can beat him at the roll of the die. Pretending that what I said makes sense, do you want to roll first or second?
Can you notice an important difference between this problem and the previous one, and explain why it matters?
You and a friend rob a mathematician's safe. Inside the safe are two envelopes. All you know is that one envelope contains n dollars, and the other contains 2*n dollars, but you don't know which envelope is which. You grab the red envelope, and your friend grabs the green one. Your friend then says, "Do you want to switch envelopes with me?"
You reason as follows: If I have the n dollar envelope and I switch, I gain n dollars. If I have the 2*n dollar envelope and switch, I lose n/2 dollars. So half the time I win n and half the time I lose n, so my expected result is 1/2 * n - 1/2 * n/2, so the switch is worth n/4 dollars on average, and you shout out "Yes, I'll take it!"
Now you have the green envelope and your friend has the red one. Suddenly he says, "I don't like the color red. Do you want to switch again?" Using the exact same argument as before, you realize that you should switch again. Under this reasoning, you now have, on average n/4+n/4 more dollars than you did at the start. But you have exactly the same envelope! How is that possible?
Moreover, it seems like you can convince yourself that you should swap with your friend again and again, without limit. This can't be correct. What is wrong with this reasoning?
I give up, show me the solution.