perlmeditation
tilly
On my home node for some time I have had an interesting
problem:
<blockquote>
Suppose I have two envelopes. All you know is that
they contain different numbers. I randomly hand you
one of them. You open it, look at it, then hand it
back. You now have sufficient information to, with
guaranteed better than even odds, correctly tell me
whether I gave you the envelope with the larger
number. How?
</blockquote>
I warn that the answer is crazier than the problem.<P>
It being Halloween I think it appropriate to publically
demonstrate that by giving the answer.<P>
<i>The trick is that you make up a number, pretend it
is the one that I still have, and that gives you just
enough information to get better than even odds,
guaranteed!</i><P>
How is <b>that</b> for sheer and utter insanity? Not
to mention being spooky!<P>
Now before I fill in details and explanations, I should
mention that this problem is not original to me. I
learned it from Laurie Snell, a well-known probability
theorist, and it has a history dating back to the 60's.
From me it found its way onto the Internet, and can be
found in such places as rec.puzzles.<P>
First details. The above answer is indeed correct, and
to have it work you need to pick your number from a
"continuous probability distribution with non-zero
density everywhere". If that confuses you, just think
of a normal distribution which is widely known as the
bell curve.<P>
Now explanations. The first is a simple algebraic
explanation. If you sit down and do the algebra, you
will find that your probability of being right turns out
to be exactly 50% plus 1/2 the probability that you pick
a number between my two. While obviously you don't know
when you pick a number between my two, you do know that
you have some chance of doing so, so you know that you
have better than even odds of being right.<P>
Some people like that explanation, some prefer to see
a picture. Well get out a piece of paper and draw a
line. Put two marks on it to represent my two numbers.
From the right mark draw an arrow left. This is the
range of numbers you need to pick in to get the answer
right if you are handed the larger number. From the
left mark draw an arrow right. This is the range of
numbers you need to pick in to get the answer right if
you are handed the smaller number.<P>
Now look closely. If you pick a number bigger than both
of mine, you have even odds of being right. Ditto if you
pick a number smaller than both of mine.<P>
But if you pick a number between mine, <b>You are right.</b>
Guaranteed.<P>
Since you always have a chance of picking a number between
my two, you always have better than even odds.<P>
See? You <i>can</i> get something for nothing! Real
information from just making something up! How much you
cannot know, but you get something! :-)
<P>
<B>UPDATE</B><BR>
This appears in the rec.puzzles FAQ as "high or low" and
their answer appears
<A HREF=http://einstein.et.tudelft.nl/~arlet/puzzles/sol.cgi/decision/high.or.low>here</A>.<P>
<B>UPDATE 2</B><BR>
I have posted Perl code at [id://39630] which shows the
argument in detail. It is a little raw, but works and shows
the key ideas in the explanation.