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

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Steve Oakley

Posts: 10
Registered: 12/17/04
Re: Beating the Odds?
Posted: Jan 30, 2013 6:06 PM
  Click to see the message monospaced in plain text Plain Text   Click to reply to this topic Reply

> On Wed, 30 Jan 2013 00:29:09 -0800, William Elliot
> <marsh@panix.com>
> 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?
>
> Of course not. Seeing one side gives you no
> information about
> what's on the other side.


Don't be so hasty.

Let a be the smaller number and b be the larger number.
Let X be the number you see and F be the cumulative
distribution function for a standard normal random
variable.

Consider this non-deterministic strategy: Guess that the
side you see is the larger number with probability F(X).

What is the probability you are correct? Condition on the side seen:

Pr[Correct]=Pr[Correct|X=a]Pr[X=a]+Pr[Correct|X=b]Pr[X=b]
Pr[Correct]=[1-F(a)]*(1/2)+F(b)*(1/2)
Pr[Correct]=(1/2)+[F(b)-F(a)]/2

Since F(b) > F(a), we have that Pr[Correct] > 1/2

Reference: Sheldon Ross, The American Statistician,
August 1994, page 267.

Note: Wikipedia attributes this approach to Thomas M.
Cover (look for the two envelopes problem).



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