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