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Path: utzoo!linus!decvax!cca!ima!ism780!jim
From: jim@ism780.UUCP (Jim Balter)
Newsgroups: net.math
Subject: Re: Re: Jim Balter's circle proof
Message-ID: <45@ism780.UUCP>
Date: Fri, 23-Sep-83 02:53:00 EDT
Article-I.D.: ism780.45
Posted: Fri Sep 23 02:53:00 1983
Date-Received: Wed, 28-Sep-83 02:05:08 EDT
Lines: 38

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(All occurrences of the erroneous "chord" in the original message have been
replaced by "arc".)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

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