Weird math

[flexagon]

I discovered this conundrum yesterday in a discussion with coworkers about a brainteaser that one of us half-heard but didn't hear the end of.

Let's say that I give you an envelope and I tell you (or you look and see) it has x dollars in it. Let's say 100. Then, I have another envelope. This envelope contains an amount that has 50% probability of having twice what you have and 50% probability of having half what you have, I tell you. Would you like to switch envelopes? The expected value of the envelope is (0.5)(200) + (0.5)(50) = $125, which is more than what you have, so of course you switch. Actually, even without doing the math you can see the situation is a bit better than double-or-nothing, so you would still say switch. As usual with these puzzles, we're assuming you are risk-neutral and not, for example, in desperate need of $90 to take your beloved pet to the vet.

So, great, we switch. You now have an envelope containing either $50 or $200 and I have an envelope containing $100. If I ask you if you want to switch again you should certainly say no. And yet, here's what baffles me: again it is entirely correct for me to say that the envelope I have has a 50% probability of containing twice what you have and a 50% probability of containing half what you have.

The English is the same. The math is reversed. And so, if I gave you a sealed envelope and told you that sentence, you should really have no idea what to do since you don't know which envelope's contents is known and which isn't.

Comments

[bluechromis.livejournal.com]

Interesting.

Reminds me of the Monty Hall problem proposed to vos Savant. Familiar with that one?

[flexagon]

The game-show "keep your door or switch?" question? Yeah, I actually learned that one in engineering risk-benefit analysis at MIT. Then I TA'd the class and... let's just say I had some convincing to do with some of the kids.

[hiddenbear.livejournal.com]

I always loved the 'Lets Make a Deal' paradox (http://www.stat.sc.edu/~west/javahtml/LetsMakeaDeal.html). Similarly difficult to wrap your head around at first, but then it does make sense in the end.

Anytime you want to play the envelope game with me though, feel free. It's win/win/win for me ($50, $100, or $200)!