🚀 go-pugleaf

     My apologies for having used the word "clearly" -- a word that is
best avoided (if something is clear you don't need to say it, and if
it isn't clear you are lying!).  However, the calculation of the first
step in the expectation with regard to the Soggies problem is not
difficult at all.  The problem is really in getting the notation
straight.

     The statement L = {0.3,0.3,0.3,0.1} means that the first three kinds
of prizes occur with frequency 30% each, but the fourth prize is only
found in 10% of the boxes.  The quantity E( L/{1,1,1,0)} ) is the
expected number of boxes that need to be purchased to get a full
set of prizes given that you have already gotten the first three.
That is, the number of boxes needed to get an example of the fourth
prize.  The standard way of finding this expectation is as the
_reciprocal_ of the probability of receiving the prize on the next box,
which is 1/(0.1), or 10.

     This is about the _only_ straightforward calculation involved in
the finding of E(L) in this case.  After wading through tons of fractions
I found that changing the distribution of prizes from {0.25,0.25,0.25,0.25}
to {0.3,0.3,0.3,0.1} raises the expected number of boxes needed to be
purchased to get a full set from 8 or so up to about 12.

     Note that what I really wanted to know here is besat expressed using
the words "or so" and "about" -- I really don't care much about the
precise fractions!  This indicates that if I wanted to carry out a bunch
of calculations of this sort I would be better off using a simulation,
which would be a lot easier anyway!  The fact that a theoretical solution
exists does _not_ mean that seeking it out is the best approach to
solving the problem, or the first method that should be tried.

     This of course has a lot of implications for our teaching of mathematics.
The elementary math curriculum is full of formal approaches to theoretical
solutions of problems, and once upon a time this was the _only_ way these
problems could be solved.  We need to make room in the curriculum for
other, more practical (and in some cases more powerful) methods as well!



   || RWW Taylor         rwtnts@rit.edu
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