lew@ihuxr.UUCP (Lew Mammel, Jr.) (09/21/83)
I don't understand Jim Balter's derivation of the circle formula either (along with Silvio Levy.) I think Silvio is being generous when he says "it seems to be sound". Recall that Jim, by his own statement, discovered the formula empirically then made up the derivation to go with it. Of the several troublesome points, I have the worst time with: "... the odds that all the points fall within a semicircle is the sum of the odds that any of the chords falls within a semicircle; since the points are equivalent, p = n * p_one_chord." This seems to be based on the false rule that the probability of a thing happening N times is N times the probability of a thing happening once. This is the second time in short succession (after Jim Stekas's "closed orbit" argument on net.physics) that I've seen a correct result supported by what appears to me to be bizarre argumentation. I find this upsetting. Incidentally, I do know the difference between "except" and "accept", I just slipped up. After I posted my offending article (which started, "I can except ...") I suddenly realized the horrible error I had committed when I was on my way out of the building. In another net.math posting I typed "M" where I wanted "M/N". That time it was actually several days later when, for some obscure reason, I suddenly realized my error. On other occasions I have posted articles with "no" for "know", "to" for "too", and "contract" for "retract". I have tried to be careful about these things ever since I was assigned to a remedial grammar class after taking a class-wide English exam my freshman year at Brown. I used "worser" in that one, among other offenses. Lew Mammel, Jr. ihuxr!lew P.S. I just found that "worser" gets through spell!
jim@ism780.UUCP (Jim Balter) (09/23/83)
Originally Silvio said my answer was wrong; I haven't seen an article in which he says either that he doesn't understand my proof or that it is sound; perhaps it is on the way. re: "... the odds that all the points fall within a semicircle is the sum of the odds that any of the arcs falls within a semicircle; since the points are equivalent, p = n * p_one_arc." This seems to be based on the false rule that the probability of a thing happening N times is N times the probability of a thing happening once. --- I carefully indicated that the N cases included all the possibilities *and* they were *mutually exclusive*, but you omit this from your quotation above: "Since the n different arcs include all the ways that the points might be formed into a semicircle, and since no two of the arcs can both fall within a semicircle (with the exception of singular cases), ..." (All occurrences of the erroneous "chord" in the original message have been replaced by "arc".) The sum of the probabilities in a complete partitioning certainly adds up to one. This is nothing at all like talking about something happening N times. It happens once, in one of N equally probable and *mutually exclusive* ways. There are N *mutually exclusive* cases; the total probability is the sum of the probabilities of these N *mutually exclusive* cases, P = P1 + P2 + ... + PN. Since P1 = P2 = ... = PN, P = N*P1. This hardly seems bizarre to me. If you want bizarre, consider the recently mentioned "paradox" where the probability of a "phenomenum" with likelihood .5 happening twice was .5+.5, with no consideration for whether the two occurrences were :-* mutually exclusive *-: Jim Balter (decvax!yale-co!ima!jim), Interactive Systems Corp --------