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Topic: Beating the Odds?
Replies: 35   Last Post: Feb 6, 2013 3:44 PM

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Dan Heyman

Posts: 168
Registered: 12/13/04
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:
>

> > On Wednesday, January 30, 2013 12:29:09 AM UTC-8, William Elliot wrote:
>
>
>

> > > There is a fair coin with a different integer on each side that you can't
>
> > > see and you have no clue how these integers were selected. The coin is
>
> > > flipped and you get to see what comes up. You must guess if that was the
>
> > > larger of the two numbers or not. Can you do so with probability > 1/2?
>
> >
>
> > Well, there was the monty hall problem years ago. Suppose there are 3
>
> > keys and you have to choose one. One of the three turns on a car's engine
>
> > where the other two do not. After you choose a key Monte takes one and
>
> > tries it in the car and it doesn't turn over the engine. He then asks
>
> > you if you'd like to change your mind. What should you do? You should
>
> > change your mind because the first key you chose had a one in three chance
>
> > of turning on the car where the remaining key has a two in three chance.
>
> >
>
> > Soo...
>
> >
>
> > Assuming the coin only has positive integers then you should guess
>
> > the number showing is the smaller integer because there are more
>
> > integers greater than x than there are less than x and this is true
>
> > for all positive integers x.
>
> >
>
> Recall, you have no clue how the integers were selected much less that
>
> both are positive or non-negative.
>
>
>
> Here's a variant. Instead of integers, the coin has
>
> 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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