Deceptive Probabilities

Here's a nice problem for you:

You are presented with three closed boxes, one of which has a prize inside. When you pick one, I will open another box for you. If that box has the prize in it, you've won. If it doesn't, and the box is empty, you can choose to open either the first box you picked, or the remaining closed box. But not both.

And so - if the opened box doesn't have the prize in it, which box should you open? The first one you picked, or the other remaining closed box? Which has the greatest probability of having the prize in it

Hint: It's deceptively easy, but it's not that obvious. Answers should be emailed to me, and the first correct one out of the hat gets a starring role (oh yes!) in the next short story I'm writing - and this is the short story I have high hopes for getting published.

No cheating, please, and people on the Culture list are excluded, since that's the place where I nicked the problem from.

31/1/00: This puzzle has already claimed its first casualties - a couple of friends who looked at this steadfastly refused to believe the (correct) answer I gave them, instead beginning to spout gibberish about quantum states and Schroedinger's Cat. Let me assure you that it is nowhere near that complex.

Results of the last puzzle