Two real numbers

Initially found from xkcd’s blog, then got a link to MathOverflow.

Suppose you are playing a game with Alice. She uses some probability distribution to pick two distinct real numbers and writes them in two envelopes. You then choose either of the envelope and opens the number inside the envelope. Is there a strategy that guarantees you a strictly larger than 50% probability to guess correctly whether the chosen real number is the greater or the lesser, for any probability distribution that Alice uses?

This is my take on the problem; feel free to comment, discuss, or argue with me! As long as the discussion is kept civil.

The answer is yes.