This is an original riddle I made, heavily inspired by the Two Envelope problem. I have my answer to this problem, but (full disclosure) even I’m not 100% sure I’ve got it right. I post it half because it seems very interesting to me, and half because I want to see if others agree with my reasoning. Enjoy.
Jack is invited to a new game show hosted by his favorite game show host, Monty Hall. To start, Monty welcomes Jack and shows him the latest in revolutionary 1970’s technology, a slot machine! But not just any slot machine. When a “betting” envelope containing cash or a check is put in front of this device and the lever is pulled, it will analyze it and dispense a “reward” envelope. 50% of the time the reward envelope will contain double the value of whatever was in the betting envelope, and the other 50% of the time it will contain only half the value of whatever was in the betting envelope. Truly a marvel of the ages.
Monty hands Jack \$100 in \$10 dollar bills and asks if he wants to take the gamble and play the machine. Having studied some basic statistics in preparation of coming here, Jack is ready to win as much money as possible and runs through some math in his head. He realizes that if he bets $10, half the time he wins \$20, the other half \$5. This gives him an expected value for each game he plays of \$12.50.
(\$20 * 0.5) + (\$5 * 0.5) = \$12.50
This machine is practically printing money!
Jack plays 10 times, winning 5 games and losing 5 games, giving him $125. Grateful that studying statistics has finally paid off, he’s about to leave when Monty announces that Jack has won the right to play the bonus game against Kirby, Jack’s hated rival from his high school days!
A cart containing 20 sealed envelopes is brought out. Monty explains each envelope contains a check with a value that is a multiple of \$2. The prizes are \$2, \$4, \$8, \$16, and so on doubling each time, up to the most valuable envelope containing \$1,048,576. Jack will chose one of these envelopes at random and let the slot machine analyze it.
The game is played as follows: Jack will receive whatever is in the original betting envelope, while Kirby will receive the reward envelope. Each of them will be allowed to open their own envelope and see what is inside, but not tell the other player. They then have the option to trade for the other person’s envelope. If both players agree to trade, the envelopes are swapped. If either of them does not want to trade, then they both just keep the envelope they started with.
Jack picks a random envelope from the 20, lets the machine analyze it, then opens it to find \$512. He remembers how he won money playing this machine earlier and eagerly asks to swap envelopes with Kirby, since he knows the reward envelope always has a better expected value.
Meanwhile, Kirby opens the reward envelope and finds \$256. Kirby reasons that there are only two possibilities, either Jack has \$128 and the machine doubled it, or Jack has \$512 and the machine cut it in half. Under the rules of this game those two scenarios are equally likely to happen. In other words, it’s a 50/50 shot for which of those two scenarios has occurred. That means that his expected value for trading is \$320, which is more than the \$256 he has.
(\$512 * 0.5) + (\$128 * 0.5) = \$320
Kirby accepts the trade, and both men walk away feeling they had gotten the better of the other.
Did either of them make a mistake?