
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 nondeterministic 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[CorrectX=a]Pr[X=a]+Pr[CorrectX=b]Pr[X=b] Pr[Correct]=[1F(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).

