I had to turn to someone as clear-headed as yourself for guidance. Note this is not analogous to the “Monty Hall” problem, where the argument for switching is correct, and not too difficult to discover. The expected value of the other envelope becomes: 50% x ($25) + 50% x ($1OO) = $62.50 leading to an expected gain of $12.50 from switching. Suppose you find $50 in the first envelope. You can assign values to the variables and not make the problem any easier. Even more disturbing are the implications of Strategy A. Nevertheless, I can’t find a flaw in the logic of Strategy B. Without ever opening the first envelope, I should switch, and then switch back (based on the same argument, defining the amount in the other envelope as y), and keep switching infinitely. The fact that Strategy B recommends switching no matter what x is makes it extremely suspicious. My sincere belief is that Strategy A is correct – it is a waste of time to switch envelopes. Similarly, there’s a 50% chance that your first choice was the envelope with the lower amount, implying a gain of x if you switch. Since there’s a 50% chance that your first choice was the envelope with the higher amount, there’s a 50% chance that you will lose by switching.
Define the amount of money you find in the first envelope you choose as x. Yet there is a powerful argument for Strategy B. Which strategy do you choose? Strategy A seems intuitively correct – there can be no gain from switching, since you picked the first envelope at random. Regardless of the contents, switch, and take the money in the other envelope. Strategy B: Pick an envelope and open it. Take the money, and proceed immediately to the nearest poker table. Strategy A: Pick an envelope and open it. You think of two possible strategies to maximize your expected profit from this situation. Since dead gamblers don’t lie, you assume that everything Mr.
You are permitted to open it, and then decide whether to keep that check, or switch envelopes and take the other check. The amount of one check is exactly twice the amount of the other check. The amount of each check is a positive real number. He informs you that inside of each envelope is a check, payable to you. As you’re strolling through the poker room at Binion’s late one night, the ghost of Johnny Moss comes up to you with two sealed envelopes. Who is that man? America’s Mad Genius - Mike Caro, of course. The only logical course of action is to ask the man with all the answers - the undisputed leader in cutting-edge thought in matters of probability.” You’ll never rest until you find the answer. Neither can some of your brightest friends. So, I said to myself: “Self, you’re a pretty bright guy – particularly when it comes to probability theory. No one has been able to give a satisfactory answer to the problem.
It has generated hours of discussion, leading recently to lower productivity here at the office. I’ve posed it to some of my brightest colleagues, both during my studies at Harvard University and at my workplace on Wall Street. Dear Mike, I’ve been confronted with a problem which I have not been able to solve. While choosing the better strategy discussed in Morell’s letter won’t directly help your poker game, this exercise might help you win indirectly by building your mental muscles. Should you switch envelopes or shouldn’t you? Think about it for two weeks, and next time I’ll respond with an analysis and an answer. This column will stretch your mind and force you to ponder.Īfter you read today’s letter from Mike Morell, try to explain the seeming paradox presented. Sometimes I open mail that’s so good I need to share it with you, even if it isn’t specifically related to poker. This article first appeared in Card Player magazine.
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