Re: Small probabilities

From: gordon brown <gbrown@euclid.colorado.edu>
Date: Sun Nov 06 2005 - 20:23:45 EST

On Sun, 6 Nov 2005, Mervin Bitikofer wrote:

> Could you find the entire Bible in pi? Hear me out on this.
>
> I'm fascinated by the strings of digits we find in a base ten
> representation of irrational numbers like pi. Consider the following as
> a very probable possibility: Any short string of digits can be found
> somewhere in pi. If for example, I wish to find four sevens in a row,
> an internet search program quickly informs me such a string is first
> found at 1589 digits after the 3. But if I want five sevens in a row, I
> have to go 162,248 digits in. Now if I understand irrational numbers
> correctly, the sequence of their digits, while determined
> mathematically, would still be, in a sense, 'random'. i.e. If there
> were infinitely repeated patterns, then it would not be an irrational
> number. So if we treat this statistically the same way we treat
> randomness, then it is never a matter of 'if' but 'where' any sequence
> of digits you wish to find may occur. We can easily work out the
> probability of finding an X length string of digits in a Y sized field.
> And that probability can never be made to go to zero as long as Y is
> larger than or equal to X. Furthermore you can make the probability of
> finding ANY desired string (/no matter how long/) arbitrarily close to
> 100% (but never quite reaching it) by simply pouring on the magnitudes
> to the search range Y.
>
> Conclusion: It is a virtual certainty of probability that the entire
> coded works of Shakespeare as well as everything else are to be found
> somewhere in pi. Okay - maybe I'm the only one fascinated by this,
> since the magnitudes of digits required to find anything interesting
> would dwarf the paltry (relatively tiny) trillions of digits now known.
>
> So if the universe was infinite in either time or space, any
> improbability becomes probable - hence the evolutionary attachment to
> steady state theories with their infinite time spans or infinite #s of
> 'parallel' universes in which so called 'randomness' can do its tricks.
> Not that I really buy any of this (or its negation either) - there are
> too many assumptions in it for either creationists or their antagonists
> to wield it effectively regarding probabilities. The conclusions
> displayed in the eventual creationist or evolutionist pedagogy are, IMO,
> more a reflection of the carefully chosen assumptions than a proof (or
> even an evidence) of a reality.
>
> --merv
>
> by the way - I'm sure my suggestion that anything can be found in
> irrational #s must be passť discussion among mathematicians, & yet I
> don't remember ever hearing such a thing proposed. Does it sound
> logical? I think there could be a formal proof done on this.
>

Although I wouldn't be surprised if any given finite string of digits
exists somewhere in pi, this is not true of just any irrational number.
Although we usually demonstrate the existence of transcendental numbers,
i.e. those that are not roots of any polynomial with integer coefficients,
by proving that the set of algebraic numbers is countable but the set of
real numbers is uncountable, there is a constructive proof of this fact
due to Liouville in 1851. The sum of the numbers 10^-j! (ten to the minus
j factorial) from j=1 to infinity is an example. The digits of its decimal
expansion are all either 0 or 1.

Gordon Brown
Department of Mathematics
University of Colorado
Boulder, CO 80309-0395
Received on Sun Nov 6 20:25:00 2005

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