Hint for to Puzzle #23: Envelopes, Stick or Change

23. I have three envelopes, into one of them I put a £20 note. I lay the envelopes out on a table in front of me and allow you to pick one envelope. You hold but do not open this envelope. I then take one of the envelopes from the table, demonstrate to you that it was empty, screw it up and throw it away. The question is would you rather stick with the envelope you have selected or exchange it for the one on the table. Why? What would be the expected value to you of the exchange?

If you don't get this you shouldn't feel bad. I've had arguments with intelligent people that, even after I've explained it, find it so counter intuitive they just sort of refuse to accept the logic.

The thing to think about is what is the probability that the twenty pound note is still on the table after you have made your selection? And does that change when one envelope is taken away? It may me helpful to pretend that the questioner knows the location of the £20, though it doesn't matter. Or imagine the same puzzle with a 1000 envelopes, 998 of them being removed.

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