|
|
Re: Beating the Odds?
Posted:
Jan 30, 2013 6:06 PM
|
|
> 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).
|
|
