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Re: Beating the Odds?
Posted:
Jan 31, 2013 8:29 PM
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On Thursday, January 31, 2013 2:17:38 PM UTC-8, peps...@gmail.com wrote:
> On Wednesday, January 30, 2013 3:14:00 PM UTC, David C. Ullrich wrote:
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> > On Wed, 30 Jan 2013 00:29:09 -0800, William Elliot <marsh@panix.com>
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> > >There is a fair coin with a different integer on each side that you can't
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> > >see and you have no clue how these integers were selected. The coin is
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> > >flipped and you get to see what comes up. You must guess if that was the
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> > >larger of the two numbers or not. Can you do so with probability > 1/2?
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> > Of course not. Seeing one side gives you no information about
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> > what's on the other side.
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> I don't completely agree with this answer. The concept of a number being selected such that "you have no clue how it was selected" doesn't translate readily into mathematics. If it's mathematics, you need to specify the procedure or specify the proability space (or set of possible probability spaces) etc.
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> Since this isn't mathematics, the best I can do is use intuition and commonsense. Surely, if one of the numbers was > 10 ^ 1000, the most intelligent guess is that the number on the reverse side is smaller.
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> Here's my algorithm: If the number < 0, I will say that the number on the current face is smallest. if the number > 100, I will say that the reverse number is smallest. Otherwise, I will answer randomly.
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> This would probably be better than 50/50.
This problem came up before in sci.math, and it is probably surprising that Ullrich's "obvious" answer is not the correct answer. The algorithm that "peps...@gmail" gave us is on the right track, but it is also not correct. For example, it is entirely possible that the numbers on the coin are chosen such that they are always both greater than 1000, in which case the given algorithm will have exactly a 50/50 chance of giving the correct answer.
I believe the problem is well specified.
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