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From: levy@princeton.UUCP
Newsgroups: net.math
Subject: Re: Largest of ten numbers (answer)
Message-ID: <20@princeton.UUCP>
Date: Tue, 30-Aug-83 23:27:10 EDT
Article-I.D.: princeto.20
Posted: Tue Aug 30 23:27:10 1983
Date-Received: Thu, 1-Sep-83 02:42:50 EDT
References: <586@ihuxr.UUCP>
Organization: Princeton University
Lines: 24

Several people came up with the right solution to this puzzle, namely letting
go M numbers out of the N and then picking the first one which is bigger than
the first M (for M=[N/e]).  Since they also provided a proof that this value
of M is optimal, I will not repeat it.

Unfortunately, I do not know of any proof that this strategy is indeed optimal
in the universe of all possible (deterministic, number-independent) strategies.
If anybody knows of one, please post it.

Ed Shepard suggests a slightly different strategy, but I did not understand
its wording.  As far as I can understand it, it consists in letting go by the
first M *positive* numbers and then picking the first number bigger than those
However this gives a probability of less than .39 for M=1, N=10, even assuming
that positive and negative numbers are equiprobable (which is not warranted
by the wording of the problem).  I should say that "choose ten real numbers
at random" doesn't make much sense because there is no uniform distribution
of probabilities over the whole real line; what I mean is "choose then real
numbers according to some distribution of probabilities, but don't tell the
guy what the distribution is".

I still think a strategy that works almost 40% of the time for this problem
is almost miraculous.

				--Silvio Levy