Estimating the Number of Jelly Beans in a JarDate: 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) |
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