## Misc Thoughts and Links

OK, I tweeted the problem, but the long form:

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?

I saw this a few weeks ago, and my answer was “Hell no.” I was wrong.

I don’t entirely trust New Scientist (they have a bit too much woo), but their article that some algae have been found to use quantum processes isn’t entirely new. (I remember reading Penrose’s book about that a decade ago … and Anathem last year).

In my general quest to read up pop (and real) psychology on decision making, I’m going through Ariely’s Predictably Irrational. I recommend it (in general, I recommend all books that make people consider their own prejudices and blind spots), but one story jumped out at me. (Paraphrasing): A psychologist had kids playing football outside his window, so he went to them and told them how much he enjoyed watching the play, and paid them each a dollar. The next day, he did the same thing, but only paid them 50 cents each. Each day, he lowered the amount he paid, until they finally said “We’re never playing here again!”.

Satisfied, the psychologist went back to his office and enjoyed the quiet. He’d turned their game into a job, (changing why they’d played from “for enjoyment” to “for money”) and then made sure it was a crappy job.

The (probably apocryphal) story reminded me why I never wanted to blog “professionally” (and why I resisted all urges to own a game store, etc etc). (I may use that excuse if I never finish my Homesteaders article).

Just because a random variable is always > 0 doesn’t imply that its expected value is > 0. It could be the case that, for every epsilon > 0, the set on which it is > epsilon is of measure 0, and I believe in this case you could have an expected value of 0.

EricFebruary 12, 2010 at 12:12 pm

I don’t see what this has to do with anything. Is it responding to a comment that was deleted? Nobody said Alice’s number picks were random or follow a particular distribution (although either could be true).

In any case, I don’t think this statement is true.

taogamingFebruary 12, 2010 at 5:29 pm

I’m replying to the claim (in the linked thread) that you can get more than a 50% chance of guessing the right number. I don’t think this has been proven by the comments.

EricFebruary 13, 2010 at 8:50 am

Meh. The linked solution assumes away part of the problem.

alexsimFebruary 13, 2010 at 8:59 pm

I find the discussion on the XKCD board far from conclusive and far from convincing. I think you should trust your first intuition on this one until you see some peer reviewed math refuting it.

SteveFebruary 18, 2010 at 9:28 am