Relay-Version: version B 2.10 5/3/83; site utzoo.UUCP
Posting-Version: version B 2.10.1 6/24/83; site nwuxd.UUCP
Path: utzoo!linus!decvax!harpo!ihnp4!nwuxd!jab
From: jab@nwuxd.UUCP (jab)
Newsgroups: net.math,net.puzzle
Subject: Re: Chain problem - clarification(?)
Message-ID: <126@nwuxd.UUCP>
Date: Sun, 26-Feb-84 18:25:48 EST
Article-I.D.: nwuxd.126
Posted: Sun Feb 26 18:25:48 1984
Date-Received: Mon, 27-Feb-84 04:22:03 EST
References: <222@pucc-i>
Organization: AT&T Technologies CSD, Lisle, Il.
Lines: 42

How long are the required lengths of chain?  Let's take an analogy:
Ask someone to choose a positive real number "at random."  What is
the probability that the chosen number is

	(1) less than 1?
	(2) less than 10?
	(3) less than 100?
	(4) less than 1000?

It's rather difficult to assign these probabilities in any objective fashion.
One thing does seem reasonable, though:  the probability density decreases
as the numbers get large.

---

Let's see. First, let's through out (2), (3), and (4), since WLOG we
can use the same argument as we'll use for (1) --- just scale the
"random number".

Now, since there are an infinite number of intervals (n, n+1], in the
range (0, infinity), the odds of you fixing "n" and then picking a
"random" positive real number in the interval (n, n+1] is almost zero.
Make sense? Let's fix "n" and try it.

I'll pick a "random" positive real number. Since there are exactly as
many real numbers in the range (n, n+1] as ((0, n] union (n+1, infinity)),
there should be a 1 in 2 chance I pick a number in the range (n, n+1].
Right?

What's wrong? I just decided earlier that it should be almost zero,
now it looks like it should be 1 in 2. Since we're working with
something that isn't finite, we seem to have wandered into an indeterminate.
I don't think that it's possible to express what you're asking for
in terms of conventional probabilities --- there's too many intervals
to play with.

It's like the friend who commented that "choosing a random integer is
dumb, since the odds of you picking one that you can pronounce in your
lifetime are almost zero."

	Jeff Bowles
	Lisle, IL