Read very, very carefully. :)
What your probability is of being correct depends on what
two numbers I have. You only can guarantee that it is
better than even, but it could be by a very small amount
indeed. There is theoretical discussion on this, and the
fact that you don't know the probability turns out
to be important. (I don't remember details though.)
However there is a variant of this problem where both you
and I pick our numbers independently out of the same
distribution. Now you can work out the probability of your
being right prior to my picking my numbers. And even though
your number has no actual information about mine, you turn
out to be right exactly 2/3 of the time. If you disbelieve
this it is easy to write a short script to test it.
Again, I got this problem from a probability theorist and
it really does work.
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