Chocolate Chips per Cookie and Poisson ProbabilitiesDate: 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! |
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