Chocolate Chips per Cookie and Poisson Probabilities

Date: 06/12/2004 at 20:40:39
From: Lina
Subject: Probability of Chocolate Chips per Cookie

A bakery makes a batch of 200 cookies in which 2000 chocolate chips 
were used.  What is the probability that a cookie picked at random
from the batch will contain at least 13 chocolate chips?

I'm not sure whether to use normal approximation here or how to go
about it if that is the correct method to use.  Taking 2000/200 = 10,
would this be the mean and the middle of the bell-curve of normal
distribution?  That would mean a 50% chance above and below that (I
think).  The number of trials is 'n' or 200. 

How would you go about tackling this question?

Date: 06/12/2004 at 23:17:46
From: Doctor Mitteldorf
Subject: Re: Probability of Chocolate Chips per Cookie

Hi Lina -

This is a problem about Poisson probabilities.  I don't know if you've 
already studied Poisson distributions, or if you have a place to look 
to read more about the subject.  

Here's how you can see a Poisson distribution coming.  When there are 
a lot of events that may happen at any time (or any place), and when 
you're counting how many of them happen at a particular time (or a 
particular place), and each event is independent of the others, then 
the result is a Poisson distribution.  This is roughly true for the 
chocolate chips; one chip being in a cookie has only a small effect on 
the probability that another chip will be in the cookie.

Once you know you have a Poisson distribution, you know a lot.  There
is only one parameter to a Poisson distribution, so if you know the 
mean, you know everything.  The form of the distribution is 

  Probability of n chips = {[e^(-x)]*(x^n)} / (n!)

In this formula, x is the average number of chips.  I think you know 
how to calculate x, from simple logic.  Once you know x, you have the 
probability for any n.

Write down the probability for n = 0 through n = 20.  Notice that 
after n = 20, the probability for greater n's is too small to worry
about.  Check that the probabilities that come from this formula for 
different values of n all add up to 1.

To avoid making these calculations by hand, a spreadsheet would be
convenient, since Excel allows you to calculate and sum Poisson
probabilities with a single function call.  If all you have is a
4-function calculator, then the Poisson sum can be quite
time-consuming, to be sure.  If you have a programmable calculator, it
may have Poisson probabilities already programmed in (TI-83, for 
example, calculates Poisson probabilities automatically).

Please write back if you have further questions on this or if anything
I said is unclear.

- Doctor Mitteldorf, The Math Forum
  http://mathforum.org/dr.math/ 

Date: 06/13/2004 at 10:47:22
From: Lina
Subject: Probability of Chocolate Chips per Cookie

Thank you for your help!  I have not studied Poisson distributions, 
but I understand your formula and it worked for my problem. I
calculated the individual probabilities and it added up correctly. 
Thanks again!