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. > > Not true: if the first seed drops near the edge, then the next seed can't fall outside the circle.
Ah, but now I can see the approach: let the first seed drop, if the probability that the next two all within the area is p, then the probability is p^2. You just calculate p for every point in the unit circle, and integrate over the area. I'll let someone else sort out the details. :-) > 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. > > It's ambiguous. The question asks about the seeds falling in "a circle", but is this circle defined beforehand or not? If it is then your solution is right. Bob -- Bob O'Hara Department of Mathematics and Statistics P.O. Box 68 (Gustaf Hällströmin katu 2b) FIN-00014 University of Helsinki Finland Telephone: +358-9-191 51479 Mobile: +358 50 599 0540 Fax: +358-9-191 51400 WWW: http://www.RNI.Helsinki.FI/~boh/ Journal of Negative Results - EEB: www.jnr-eeb.org
