October 15, 2005

Eavesdropping for nerds.

What hot gossip were you overhearing when you just couldn't resist breaking in and joining a private conversation? Or perhaps you were sitting near people who were discussing a logic problem, maybe this one:
Someone has prepared two envelopes containing money. One contains twice as much money as the other. You have decided to pick one envelope, but then the following argument occurs to you: Suppose my chosen envelope contains $X, then the other envelope either contains $X/2 or $2X. Both cases are equally likely, so my expectation if I take the other envelope is .5 * $X/2 + .5 * $2X = $1.25X, which is higher than my current $X, so I should change my mind and take the other envelope. But then I can apply the argument all over again. Something is wrong here! Where did I go wrong? In a variant of this problem, you are allowed to peek into the envelope you chose before finally settling on it. Suppose that when you peek you see $100. Should you switch now?
Would you become so intrigued that you would ask to join the conversation?

(Here's the solution to the problem.)

ADDED: Let me be clear: I was not the eavesdropper! I was one of the original conversationalists, but I was not the person who introduced the topic. Of the three persons described here, I was the one demonstrating the least nerd cred.

9 comments:

whit said...

Not on a Saturday afternoon.

erp said...

It wasn't a logic problem they were getting wrong, but hurricane Katrina and the disaster that is New Orleans.

Next to our table in a fairly upscale restaurant on a Florida beach was a table of six well dressed and well spoken people having lunch and discussing the horrors of hurricanes in general and Katrina in particular. The problem was that although they faithfully repeated what they learned from the media, they really had no idea of the facts at all. We were hard put to stay in our own seats and eat our lunch quietly.

What they believe will be entered into the history books as fact and I wonder if anything can be done to correct the record

John Thacker said...

I think about the first half of that explanation is quite misleading, all the comments about it being a constant and what not.

The statement of the argument is presuming the existence of a uniform probability distribution over all positive numbers. (It's really a nicely hidden assumption, too.) Since that cannot exist, the problem is assuming something false and thus it's not surprise when it leads to some other contradiction or impossible situation. Once you assume some actually possible prior distribution, perhaps an upper limit on how much money can be in an envelope, the puzzle disappears. (Although in that case whether it makes sense to switch depends on what you believe the odds are of different amounts in the envelopes.)

It is true that if your chosen envelope contains $X, then the other envelope contains either $X/2 or $2X. But it does not follow that either case is equally likely regardless of X under any real probability distribution.

amba said...

nerd cred! That is tantalizingly close to being an anagram, or a rhyme, or a palindrome, or something. It isn't anything, but it looks like it should be.

chuck b. said...

I overheard someone telling her companion about her Japanese blood grass turning green. Without skipping a beat, I turned around and interjected*, "You have to rip rip up all the green clumps! They will never turn red and they will take over!"

At that moment, I felt like the biggest dork on the planet. Not really a geek, because gardening is more dorky than geeky, if either. At least they were interested in what I had to say.

*"ejaculated" would be the best word, but it would also arouse the twitterers which is too bad, cuz it's too good a word to get such limited use.

chuck b. said...

Oh, and I just read the whole post...Marilyn Vos Savant ran this problem in her Parade column and advised picking the second-choice envelope. I know this because I read the column where she described the firestorm her answer kicked up.

Reading Parade...and telling the world about it on Althouse. That's geeky.

Ann Althouse said...

Chuck: I think you're thinking of the Monty Hall problem, which has three doors and involves the actions of a person who knows which door is the good one.

vbspurs said...

Question to everyone:

Are you one of those people with whom perfect strangers like to strike up a conversation?

Say when you're standing in bank queues, or at the supermarket, or...well anywhere really.

I am.

It's a curse I tells ya.

Cheers,
Victoria

Chris said...

That link didn't work for me, but http://www.rec-puzzles.org/ sol.pl/decision/envelope did.