
Re: Beating the Odds?
Posted:
Feb 1, 2013 2:32 PM


In article <Pine.NEB.4.64.1301300026190.3747@panix3.panix.com>, 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?
Yes, provided you have a random variable at your disposal, say a standard normal random variable X. If the number showing on the coin is less than X, then guess that the number on the other side of the coin is the larger of the two. If the number showing is greater than or equal to X, then guess that the number showing is the larger. Your chance of being correct is
(1/2)[1Phi(x)] + (1/2)Phi(y) = 1/2 + (1/2)[Phi(y) Phi(x)] > 1/2.
Here Phi is the standard normal distribution function, and x < y are the two numbers on the coin.  J.

