Estimating the Number of Jelly Beans in a Jar

Date: 8 Feb 1995 20:36:06 -0500
From: Marie Feigel
Subject: Science Project Help

Dear Dr. Math,

     I'm doing my fifth grade project on probability.  I need help
naming/finding the reason why the average of the guesses of how 
many things are in a jar is a good prediction of the exact number 
of things in the jar.  My mom thinks that it is the Lincoln Sampling 
Method, but she is not sure.  HELP!

                                                Thanks,
                                                Paul Feigel
                                                Houston Elem.
                                                Phila

Date: 14 Feb 1995 14:22:47 -0500
From: Dr. Ethan
Subject: Re: Science Project Help

Hey Paul,

        Sorry it has taken so long to get back to you.  I have been 
thinking about this a lot but I guess I don't have much to say.  I'm 
not sure the notion of "good prediction" is entirely clear.  Do you 
mean that the average will have a higher probability of being right 
than any other guess? (This seems obvious but would require some 
proof), or do you mean that the probability that the average is 
correct goes to one.  This second statement is much more powerful, 
and I don't think that it is true unless you have some way of 
guaranteeing that the guess will be distributed in some random 
fashion.

        If none of this makes sense please write back to us and we 
will try to be of more help.  

                Ethan Doctor On Call

PS.  I looked in a couple of probability texts but could not locate 
the Lincoln sampling method.  I will try to talk to a professor in the 
next couple of days.

Date: 15 Feb 1995 20:49:18 -0500
From: Anonymous
Subject: Re: Science Project Help

Paul,

        The answer to your question is a little tricky because
the average of the guesses isn't necessarily a very good predictor,
though it may be a better predictor than just one of the guesses.
I think there's as much psychology involved in the answer as there
is math, so I'll explain it from that point of view and see if
you're satisfied with the answer.

        First off, why is there any inaccuracy at all?  Why don't
all the people guess the right answer?  Well, people aren't that
accurate at guessing how many things there are in a jar, and the
inaccuracies come from two sources.  One is that the method they
use to estimate might be flawed.  For example, they may have seen
100 jellybeans in a jar once, and the current jar looks twice as
big so they guess 200, not realizing that a jar that looks twice
as big is probably more than twice as big.  (This is because
doubling the length, width and height of a jar multiples the
volume by 8 and people often don't take into account the fact
that all 3 dimensions have been doubled.)  Anyone making this
kind of mistake will guess a number that is too low, so averaging
guesses from a bunch of people like this will result in an
answer that's just as much too low.  Other mistakes of this sort
that people make are to think that a large number of things (say 100,000)
"just couldn't" fit in a jar, also resulting in guesses that are
systematically too low, or that a number like a trillion is just
a bit bigger than a billion (since they're both big numbers, and
one has just a few more zeros than the other), which results in
guesses systematically WAY too high.  You might think these
systematic too-high errors would cancel out the systematic too-low
errors, but I think it's likely that most of the systematic errors
will be the same way (people think alike) or that the too-high
ones will dwarf the too-low ones when you average.

        Secondly there are errors resulting from imprecision, namely,
even if you have a good way of measuring such as weighing them,
your scale won't be completely accurate.  It will be off by a bit,
in what is probably a random, unbiased amount, and the errors
made by different people will probably be "independent", that is,
unrelated.  There is a mathematical fact, that if you have a lot
of independent and unbiased errors, the average of these
errors will tend to zero as the sample gets large, meaning that
the average of the people's estimates will tend toward the
true value.  This fact is usually called the Strong Law of Large Numbers,
but sometimes goes by other names.  To tell the truth, I've never
heard of the Lincoln Sampling method, so I don't know whether or not
that is another name for this.

        The upshot of this is that averaging reduces the error
due to imprecision, but doesn't do much about the error due to
faulty method or intuition.  (When I refer to faulty method in
the second paragraph, I mean to include the "wild-guess" method,
which usually means that someone picks a number they feel is
a good guess without knowing why; usually this means they have
some subconscious method of coming up with a plausible estimate,
which is subject to the same flaws as if they had consciously
used one of the methods I mentioned.)

        I hope this helps and I apologize for the delay in answering.

Dr. Math (Robin Pemantle)