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Envelopes Paradox

Here's a little puzzle that was thrown at us in a probability lecture back in uni. Sad to say it took me quite a while to figure out -- maybe you'll have more luck.

The setting is the traditional gameshow environment, and your aim is to get as much money as possible. You've just been offered a choice between two envelopes. They both contain money and you're informed that one of the envelopes contains twice the money that's in the other.

With nothing else to choose between them, you pick one at random. The host asks whether you're sure, or whether you want to switch.

Thinking About It

Being a mathematician at heart, you reason thus. Call the amount of money in the envelope you did pick x pounds. The other envelope therefore contains either 2x or x/2 pounds, and these two possibilities are equally likely. If you switch, your expected reward is the mean of these two outcomes, which is 5x/4. So there's a net gain in switching, and you choose to switch.

Or Not

The above argument is nonsense -- there's no reason to switch. The question is, can you spot the flaw?

This puzzle goes hand in hand with the Monty Hall Paradox.