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Re: Beating the Odds?
Posted:
Feb 1, 2013 10:27 AM
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On Thursday, January 31, 2013 3:23:35 AM UTC-5, William Elliot wrote:
> On Wed, 30 Jan 2013, forbisgaryg@gmail.com wrote:
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> > On Wednesday, January 30, 2013 12:29:09 AM UTC-8, William Elliot wrote:
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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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> >
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> > Well, there was the monty hall problem years ago. Suppose there are 3
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> > keys and you have to choose one. One of the three turns on a car's engine
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> > where the other two do not. After you choose a key Monte takes one and
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> > tries it in the car and it doesn't turn over the engine. He then asks
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> > you if you'd like to change your mind. What should you do? You should
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> > change your mind because the first key you chose had a one in three chance
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> > of turning on the car where the remaining key has a two in three chance.
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> >
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> > Soo...
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> >
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> > Assuming the coin only has positive integers then you should guess
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> > the number showing is the smaller integer because there are more
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> > integers greater than x than there are less than x and this is true
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> > for all positive integers x.
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> >
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> Recall, you have no clue how the integers were selected much less that
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> both are positive or non-negative.
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> Here's a variant. Instead of integers, the coin has
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> a positive rational on each side.
It's even easier if there is a real number on each side. A method to get
better than 50-50 odds appeared in a paper many years ago, written by a Stanford professor named Cover (I think). It's very simple, and the proof that it works is a one-liner.
Here's a hint: You want to pick the side you see if the number on it is "large". How can you decide if a number is "large"?
Note: The range of the possible numbers can be (-oo,oo) or any finite interval, e.g. [0,1], and the basic idea is the same.
Also, the way the numbers on the coin are chosen is irrelevant.
Dan Heyman
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