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The Backgammon Solution
Posted By: Casper Van der Tak
Date: Sunday, 16 October 2016, at 10:27 a.m.
In Response To: The Backgammon Solution (Rick Janowski)
Yes, but there is a difference between knowing the amount or talking about an unknown amount. Say for example 50% of the time it is 10002000 and 50% of the time it is 20004000, and that in both cases you have equal chances to take the lower or the higher amount.
Purely random draw will get you 2500
If you get to look at the envelope, you keep it if you get 4000, and you switch if you get 1000. You also switch when you get 2000. You get 2750 on average.
Now if you switch without looking, and call the earlier selected amount A, then:
25% of the time you drew 1000, switch to 2000, and gain A = 1000 from switch
25% of the time you drew 2000, switch to 4000, and gain A = 2000 from switch
25% of the time you drew 2000, switch to 1000, and lose 0.5A = 1000 from switch
25% of the time you drew 4000, switch to 2000, and lose 0.5A = 2000 from switch
In terms of money gains and losses, you don't gain from the switch. In terms of "A", you seem to gain an expected amount of a quarter of A; however, in the scenarios where you gain, the value of A is half that of the scenarios where you lose, so there is no actual gain (which is logical, because you did not get any additional information that could help decisionmaking).

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