> On Wed, 30 Jan 2013 00:29:09 -0800, William Elliot > <firstname.lastname@example.org> > 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: