Ted writes:

> Wirt writes . . .
>  >The binomial theorem explains all.
>
>  >So long as you have a "barb of selection" (or a reified Maxwellian
Daemon)
>  >operation, observing and testing every trial, exceedingly improbable
events
>  >become essentially inevitable given enough trials.
>
>  Hogwash.
>
>  In the case of two trials at 1/2 chance, that is indeed
>
>    1 - (1/2)^2 = 3/4 = 75%
>
>  However, in the case of three trials at a chance of 1/3 that the event
>  happens, the formula is
>
>    1 - (1 - 1/3)^3 = 19/27 (about 70.4%)
>
>  In general we have this.  Given the probability of 1/n that an event
occurs,
>  the probability that it does not occur is 1-1/n.  The probability that it
>  not occur in n trials is then (1 - 1/n)^n and the probability that it
occurs
>  least once in n trials is 1 - (1 - 1/n)^n.  If we examine this as n gets
>  large, we get
>
>                       1  n                    1  n        1
>    lim      1 - (1 - ---)   = 1 - lim   (1 - ---)  = 1 - ---
>    n->oo              n           n->oo       n           e
>
>  (for those without a fixed-width font, that's 1-1/e or about 63.2%.

That's exactly correct. There are a couple of morals in there. One is never
write this kind of stuff while you're enormously busy doing something else
(and so tired you can't see). Another is never do mathematics solely in your
head. Pen never touched paper in my previous posting. If it had, I would have
immediately seen the error of my ways. The equation Ted posts above is one of
the most common in all of physics. A third moral is that there is always
great value in peer review. Errors creep into everything, and the only way to
expunge them is to have any bit of work reviewed by a few others.

But Ted's next statement could lead you philosophically astray:

>  So the
>  chance of something with 1/n probability happening at least once in n
trials
>  is somewhere between 75% and 63% which is still a long way from a
certainty.

To that, Steve writes exactly the right response:

> The situation described is more
>  like "the probability is 1 in seventeen quintillion, but you get to take
>  243 septillion shots at it".

While the probability of an extremely rare event occurring is (1 - 1/n)^xn,
if we let x represent some multiple of n, calculating the probability of this
event occurring is simple enough to calculate that you can *do* it in your
head: 1 - (.37)^x. If x is as small as 10, then the probability of the event
rises from 63% to 99.995%.

In physics, it's only half-kiddingly said that there are only three answers:
0, 1 and infinity. In this case, the answer is 1. If some form of "barb of
selection" exists, which might be some form of physical condensate or the
differential survival of some favored but exceedingly rare genetical
combination, no matter how rare that favored circumstance might be, its
existence becomes as near certainty as you would care to imagine, given that
you can take a sufficient number of shots at it.

Thus, the moral of the first posting remains, regardless of the arithmetic
errors: exceedingly low-probability universes are not at all necessarily
low-probability. Indeed, given the presence of a barb of selection, these
universes may well be inevitable.

Wirt Atmar

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