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From: mason@utcsrgv.UUCP (Dave Mason)
Newsgroups: net.math,net.misc,net.rec.bridge
Subject: Re: simple (?) statistics problem solved
Message-ID: <2034@utcsrgv.UUCP>
Date: Sat, 20-Aug-83 04:58:03 EDT
Article-I.D.: utcsrgv.2034
Posted: Sat Aug 20 04:58:03 1983
Date-Received: Sat, 20-Aug-83 05:47:33 EDT
References: <657@vax2.UUCP>
Organization: CSRG, University of Toronto
Lines: 31

Also ref 1996@umcp-cs.UUCP & 567@ihuxr.UUCP

Lew suggested looking at 3 cabinets with: 10s; 9s1g; 10g coins.
By changing the number of coins so that the number of gold coins does not
match the number of cabinets the problem becomes easier (and less interesting).

Think of labelling all the coins with 1, 2 or 3 for the cabinet in which they
are found, then dump all 30 coins in a bowl.  Take a coin out of the bowl.
You immediately see that it is gold.  What is the chance that it is has a 1
marked on it? what is the chance it has a 2? 3?

I submit that the probabilities are 0,1/11 and 10/11: there are 11 gold coins,
and only one of them has a 2.  We still haven't looked to see what it was,
but we're going to predict what is most likely to be on the next gold coin we
get out. In fact the probability is 10/11 that again it will be a 3.
(there is a 1/11 chance that the first one was a 2. If it was then the next
one will be a 3 (there's only one 2) ie with a prob of 1 so the overall
probability so far is 1/11x1=1/11.  But there was a 10/11 chance that the first
was a 3.  If it was then there is a 9/10 chance that the next will be a 3 so
this half of the possibilities contributes an overall prob of 10/11x9/10=9/11.
The total probability of the second gold being a 3 is the sum of these two:
1/11+9/11=10/11.  The math gets a bit messy, but the probability stays the
same for all 11 gold coins we draw.)

The result of this is that if we get a gold coin when we first walk into the
room we should keep drawing from that cabinet as long as we get gold, and
switch as soon as we get silver. (maybe this was the original question)

 -- Gandalf's flunky Hobbit --   Dave Mason, U. Toronto CSRG,
        {cornell,watmath,ihnp4,floyd,allegra,utzoo,uw-beaver}!utcsrgv!mason
     or {cwruecmp,duke,linus,lsuc,research}!utzoo!utcsrgv!mason   (UUCP)