So, I said to myself: "Self, you're a pretty bright guy -
particularly when it comes to probability theory. Yet you can't solve this
problem. Neither can some of your brightest friends. You'll never rest until you
find the answer. 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."
Who is that man? America's Mad Genius -- Mike Caro, of course. [You can see why
I like this letter, right? - MC] So, Mike, here's the problem:
Johnny Moss' envelopes. 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. He informs you that inside of each envelope is a check, payable to
you. [We have to stretch our imagination here to assume the legendary Johnny
Moss is offering checks and not cash. - MC] The amount of each check is a
positive real number. The amount of one check is exactly twice the amount of the
other check. You may choose either envelope. You are permitted to open it, and
then decide whether to keep that check, or switch envelopes and take the other
check. Since dead gamblers don't lie [Morell adds a footnote explaining that
this phrase comes from the book Caro on Gambling. - MC], you assume that
everything Mr. Moss has told you is true. You think of two possible strategies
to maximize your expected profit from this situation.
Strategy A: Pick an envelope and open it. Take the money, and proceed
immediately to the nearest poker table. Don't waste time switching.
Strategy B: Pick an envelope and open it. Regardless of the contents, switch,
and take the money in the other envelope. Then play poker.
Pick a strategy. 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. Yet there is a powerful argument for Strategy B. Define the
amount of money you find in the first envelope you choose as x. 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 [half] by switching. 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 [you'll double].
[I have omitted Morell's carefully presented formulae defining the mathematical
expectations of the two choices, because they go beyond what most readers want
to wrestle with, and we can ponder this seeming paradox without them. - MC]
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