RE: Small probabilities

From: Bill Hamilton <williamehamiltonjr@yahoo.com>
Date: Sun Nov 13 2005 - 09:39:37 EST

This problem has turned out to be quite a bit more interesting than I had
expected. Glenn's answers, and those from others point out the practical
difficulty of actually performing what I proposed as a thought experiment.
However, Glenn's digital precision answer inspired me to consider whether I
could achieve a lesser objective: produce a random variable whose probability
of occurrence is < 10^-150 -- the value which Dembski eliminates chance. The
value of RAND_MAX -- the maximum value a random integer can take in a given
environment -- is 2^31-1 for Mac OS X (probably also for Windows, but I looked
it up at home) which is equal to 10^9.331929865. (say 10^9). A call to ran1,
the numerical recipes uniform random number generator takes a few microseconds
(I haven't timed it, but I have been running some simulations that make
thousands of calls to it, and the calls seem to add very little delay. So call
ran1 17 times and concatenate the results, considering the result to be a
random number between 0 and 1 and you have produced a random variable whose
probability is < 10^-150.

Of course this ignores the problem of the periodicity of random number
generators, but there are ways of getting around that.

--- Glenn Morton <glennmorton@entouch.net> wrote:

> One can believe in the Platonic line (or the Pythagorean line) all he wants.
> The reality was that there is nothing in existence which has a infinitely
> fine point with which to pick a point. And there is not an infinite time in
> which to write down the digits required to specify any point. Let's say we
> give you the fastest computer on earth to randomly pick a point on the
> line-Let's let it go for a million years spitting out numbers after the
> decimal point. When it finally runs out of resources, it stops at a number
> and the result is a quantization of the line below the size of the place
> held by the last digit. So, lacking the time, also quantizes the choice
> and thus the probability is NOT zero for whatever point is humanly possible
> to pick.
>
>
>
> Sure this is an argument from practicality and reality rather than
> mathematics, but I would argue that only if it is actually possible to pick
> any point whatsoever is the real probability really zero for each point in
> the line. To illustrate this Consider the output of the computer for that
> random selection. It looks like:
>
>
>
> .49292238460049- - -365
>
>
>
> The five is the last digit spit out after the computer has run out of time,
> resources, electricity, or the end of the universe happens just after the 5
> is printed out. That then quantizes the line at
>
>
>
> .00000000000000- - -001
>
>
>
> Since that is a finite number the probability for points to be selected is
> .00000000000000- - -001 for the points that lie at this quantization and
> zero for points in between this quantization. Those points in between can
> not possibly be picked and thus they are the points which have zero
> probability of being picked.
>
>
>
>
>
> _____
>
> From: asa-owner@lists.calvin.edu [mailto:asa-owner@lists.calvin.edu] On
> Behalf Of Alexanian, Moorad
> Sent: Sunday, November 06, 2005 11:00 AM
> To: Glenn Morton; 'Bill Hamilton'
> Cc: asa@calvin.edu
> Subject: RE: Small probabilities
>
>
>
> There are indeed an infinite number of points in a line and so strictly
> speaking the probability is zero to find any particular point. The latter is
> mathematics and the real question has to be based on experiments. One always
> deals with a large, but finite number of outcomes---a die with a large
> number of sides, say. Note also that when one measures lengths---which,
> presumably, have an infinite number of mathematical points---one uses
> smaller lengths that also have an infinite number of mathematical points.
> Any measuring device deals with finite lengths. One has to distinguish
> mathematics that deal with infinities with reality, which deals with
> finiteness.
>
>
>
> Moorad
>
>
>
> _____
>
> From: Glenn Morton
> Sent: Sun 11/6/2005 10:32 AM
> To: 'Bill Hamilton'
> Cc: asa@calvin.edu
> Subject: RE: Small probabilities
>
> > -----Original Message-----
> > From: asa-owner@lists.calvin.edu [mailto:asa-owner@lists.calvin.edu] On
> > Behalf Of Bill Hamilton
> > Sent: Sunday, November 06, 2005 8:01 AM
>
>
> > I read Dembski's response to Henry Morris
> > (http://www.calvin.edu/archive/asa/200510/0514.html)
> > and noted that it raised an old issue I've harped on before: that you can
> > specify a probability below which chance is eliminated. There is a
> > counterexample given (among other places) in Davenport and Root's book
> > "Random
> > Signals and Noise" (McGraw Hill, probably sometime in the early 60's) that
> > goes
> > like this:
> > Draw a line 1 inch long. Randomly pick a single point on that line. The
> > probability of picking any point on the line is identically zero. Yet a
> > point
> > is picked. Am I missing something?

Bill Hamilton
William E. Hamilton, Jr., Ph.D.
586.986.1474 (work) 248.652.4148 (home) 248.303.8651 (mobile)
"...If God is for us, who is against us?" Rom 8:31

                
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Received on Sun Nov 13 09:42:05 2005

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