My apologies if this is too elementary, but it's been years since I heard of this paradox and I've never heard a satisfactory explanation. I've already tried it on my fair share of math Ph.D.'s, and some of them postulate that something deep is going on.
The Problem:
You are on a game show. The host has chosen two (integral and distinct) numbers and has hidden them behind doors A and B. He allows you to open one of the doors, thus revealing one of the numbers. Then, he asks you: is the number behind the other door bigger or smaller than the number you have revealed? Your task is to answer this question correctly with probability strictly greater than one half.
The Solution:
Before opening any doors, you choose a number $r$ at random using any continuous probability distribution of your choice. To simplify the analysis, you repeat until $r$ is non-integral. Then you open either door (choosing uniformly at random) to reveal a number $x$. If $r < x$, then you guess that the hidden number $y$ is also smaller than $x$; otherwise you guess that $y$ is greater than $x$.
Why is this a winning strategy? There are three cases:
1) $r$ is less than $x$ and $y$. In this case, you guess "smaller" and win the game if $x > y$. Because variables $x$ and $y$ were assigned to the hidden numbers uniformly at random, $P(x > y) = 1/2$. Thus, in this case you win with probability one half.
2) $r$ is greater than $x$ and $y$. By a symmetric argument to (1), you guess "larger" and win with probability one half.
3) $r$ is between $x$ and $y$. In this case, you guess "larger" if $x < y$ and "smaller" if $x > y$ -- that is, you always win the game.
Case 3 occurs with a finite non-zero probability $\epsilon$, equivalent to the integral of your probability distribution between $x$ and $y$. Averaging over all the cases, your chance of winning is $(1+\epsilon)/2$, which is strictly greater than half.
The Paradox:
Given that the original numbers were chosen "arbitrarily" (i.e., without using any given distribution), it seems impossible to know anything about the relation between one number and the other. Yet, the proof seems sound. I have some thoughts as to the culprit, but nothing completely satisfying.
Insightful members, could you please help me out with this one? Thanks!