

Randall Munroe's Math Puzzle - BIackSwan

He posted it on his G+ wall. (https://plus.google.com/111588569124648292310/posts)<p>I pick two real numbers through some process unknown to you. It might be random and it might not. Maybe I always pick "3" and "100". Maybe I roll two dice. Maybe I write C code by mashing a keyboard until it compiles and prints two numbers (or produces Windows ME). Maybe I always use 0 for the first number, and for the second I call my aunt and ask her for a negative real number, which I multiply by the estimated number of protons in the universe. (At this point, my aunt is used to that kind of call from me.)<p>I put these two numbers on slips of paper and put them in two envelopes. I thoroughly shuffle the envelopes, and then you choose one via a fair coin toss. You open it and look at the number. You are now given the option (as in the infamous but very different Monty Hall problem) of switching to the other envelope.<p>Your goal is to pick the envelope with the higher number. Can you come up with a strategy that guarantees you a better-than-even chance of winning?<p>It has to do better than 50% no matter how I picked the numbers--if your strategy includes the rule "switch if the number is a 2", it's wrong, because I could always be picking 2 and negative 100, and in that scenario your strategy will fail at least 50% of the time. This means your strategy must work even if I have guessed what your strategy is and am cherry-picking numbers specifically to defeat it.<p>I cannot emphasize enough that you do not know anything about my process. Not only do you not know the numbers, you don't know which random distribution I am picking them from—or whether it's even a random distribution at all—and there is no deductive basis for estimating the odds that I'm using any particular method. (This gets at what is sometimes called the difference between risk and uncertainty.)<p>I'll post again tomorrow with the correct answer and a bit more about the puzzle.
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ColinWright
Firstly, you don't have to shuffle the envelopes, and you don't have to
specify that I must use a fair coin to choose one. The problem as stated is
unnecessarily over-specified.

Secondly, you say:

    
    
      > Can you come up with a strategy that guarantees
      > you a better-than-even chance of winning?
    

It should be pointed out that this should be a fixed strategy that will be
used on every occasion we play this game. The challenge is to find a strategy
which will, independently and on every occasion we play, have, on that
occasion, a chance of winning that is strictly greater than 50%.

Finally, this is well-known (in some circles) and controversial. I remember
getting into a protracted discussion (probably around 100 emails) with someone
claiming that the stated strategy that achieves better than 50% was wrong, and
that it was in fact still exactly 50%. It took a very, _very_ long time to
convince them that they were calcualting the wrong thing.

So for your sake, I hope you get the explanation crystal clear, and avoid
_all_ the pitfalls. Otherwise you will be hounded.

Very brave.

 _Added in edit: I've just gone and looked at the comment thread there, and
now I'm depressed. I should know better than to look at people trying to talk
about math and probability, even a sub-group such as comment on XKCD blag
posts._

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BIackSwan
Lets see the solution he puts up.

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ColinWright
I'm pretty sure there's only one solution (I have no proof of that) and I'm
pretty sure he'll put up the solution I know. There will then be a deluge of
replies from people who either don't understand the problem, think it's a
different problem, or don't understand the solution.

Already in the comments we see the solution, followed by many _many_ posts
from people who really just don't understand it.

The cynic in me says it's a great way to generate traffic, but it's a really
poor way to create understanding. It's almost a math equivalent to FUD.

If you want to know my solution, email me. Address is in my profile. Although
the lack of upvotes suggests that no one is that interested, I won't post it
here and spoil the fun in case someone wants to think about it first.

In case you're interested, I've uploaded a statement of the problem (slightly
altered to make it a little more general) along with a solution. The MD5 of
the file is 0a600b1f7f26c44854b8a8118f5491a1 - so if you ask later and
download the file you can check that I've most likely not changed it.

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SamReidHughes
Well obviously you just keep playing until you've won more than you've lost,
and then you stop playing.

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gujk
For the record, the solution assumes the existence of an infinitesimal real
number larger than zero, or a finite bound on the adversaries' range of
choices. The larger his range, the smaller your chances of beating 50-50, with
no limit on how small your chances may be.

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gujk
Randall Monroe posted the solution ages ago on his blog. Is he syndicating
himself on G+?

