# Re: Small probabilities

From: Iain Strachan <igd.strachan@gmail.com>
Date: Sun Nov 13 2005 - 16:37:26 EST

A simpler lo-tech solution than calling a random number generator is to toss
a coin 500 times, or throw a die 200 times and concatenate the values. Both
of these will generate events with probability < 10^-150. But as I said
earlier that's no big deal. It's a big deal when a pre-specified event of
very small probability occurs because then there are a limited number of
outcomes, as opposed to a vast number. Randy's example of getting the same
deal in a pack of cards twice in a row is an example of this. Or if I tossed
a coin 500 times and you tossed a coin 500 times and we compared notes later
and found we'd got exactly the same sequence. In the case of a single 500
coin toss sequence, there are 10^150 possible outcomes, each with
probability 10^-150, so there is no big deal. For the second coin-toss
sequence to match, there is only ONE outcome that achieves this, so it then
becomes remarkable that this specified event of probability 1e-150 has
occurred.

Iain

On 11/13/05, Bill Hamilton <williamehamiltonjr@yahoo.com> wrote:
>
> 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.
> >
> >
> >
> >
> >
> >
> > _____
> >
> > 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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```--
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There are 3 types of people in the world.
Those who can count and those who can't.
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```
Received on Sun Nov 13 16:41:24 2005

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