Wirt, You are right about your concern with the question. After all, it should be,
Given that three seeds fall randomly in a one meter diameter circle, what is the probability that all three seeds will be within one meter of each other? There should be no "first" seed involved in the calculations of probability, since the question asks about three seeds that fall at random. After all, it could fall near the edge, and it would not be the center of a 0.25 pi circle, because > half of its circle would fall outside the larger circle. Once you define the question by using the first seed, then you also have to state where it fell in order to calculate the remaining probabilities. Imagine that the first seed fell on the edge, but on the inside edge. Then its "circle's" area is less than 0.125 pi. So, the probability for each of the next two seeds would be > 0.875 to fall outside the first semicircle (or < 0.125^2 = 0.0156 again). But, that is a conditional probability, that ignored the probability involved with where the first seed fell. Jim Wirt Atmar wrote: > Regarding my earlier answer to the question, two people have written me and > said that I misread the question. That's entirely possible: > > >> This assumes the smaller circle is defined within the bigger. If the >> question concerns *any* smaller circle within the bigger one, then the >> > first > >> seed could fall anywhere. >> > > Probabilities are always calculated on the basis of what specifies success or > failure. I was presuming that the smaller circle was pre-specified, lying > somewhere within the larger one. If that were the case, my previous answer > would > be correct. > > However the way that others are reading the question is that the first seed > specifies the center the smaller circle, essentially the same if the question > were more along the lines of "what's the probability of getting a duplicate?" > In that case, the first draw species what a duplicate would be. In this > alternate interpretation, the drop of the first seed doesn't count. Rather, > its > position will be used to specify the center of the circle. If that is the > case, > then the probability of getting the next two seeds within that first > seed-specified circle is 1/4 * 1/4 = 0.0625. > > However, I am still a little reluctant to endorse that interpretation simply > because the original question asked: "What is the probability that all three > will be clustered within a circle of one-half meter radius?" It's the "all > three" part of the question that gives me pause. > > Wirt Atmar > > -- ------------------------------------- James J. Roper, Ph.D. Universidade Federal do Paraná Depto. de Zoologia Caixa Postal 19020 81531-990 Curitiba, Paraná, Brasil ===================================== E-mail: [EMAIL PROTECTED] Phone/Fone/Teléfono: 55 41 33611764 celular: 55 41 99870543 e-fax: 1-206-202-0173 (in the USA) ===================================== Ecologia e Conservação na UFPR http://www.bio.ufpr.br/ecologia/ ------------------------------------- http://jjroper.sites.uol.com.br
