A math problem

So Chris was reading the xkcd blog and found this math problem, which of course he had to tell me about… Now we’re both irritable and annoyed trying to figure it out. Here’s the problem:

Alice secretly picks two different real numbers by an unknown process and puts them in two (abstract) envelopes.  Bob chooses one of the two envelopes randomly (with a fair coin toss), and shows you the number in that envelope.  You must now guess whether the number in the other, closed envelope is larger or smaller than the one you’ve seen.

Is there a strategy which gives you a better than 50% chance of guessing correctly, no matter what procedure Alice used to pick her numbers?

And by the way, yes there is a strategy that gives you a better than 50% chance of guessing correctly… But the question is, can you figure out what it is?

We’ve both been dwelling on this since Friday morning, having heated arguments about dividing infinity at zero, if it’s possible to assume negative infinity (<0) is equal to positive infinity (>0), if the first number is a negative number (like -12) then would it be an advantage to say the next number will be higher because the probability of that is positive infinity + 11, but of course infitinity + any number is still equal to infinity. But couldn’t you could divide infinity in half at any number (like -100) cause then both “halves” would still be infinity. Argh!

Now we’re considering conditional statistics, like the famous Monty Hall problem. (Another excellent mind twister if you haven’t seen it before)

If you want the answer you can click on the “xkcd blog” link above and click to view the comments. The answer is supposed to be in the first comment. We haven’t looked, we have a 3 day weekend and we’re determined to use it! So don’t tell us the answer if you read it!

Good luck!

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