(Ted, thinking about other things to this point, suddenly realizes he's being
baited)

Wirt writes . . .
>The binomial theorem explains all. What's the probability of a really
>beneficial event occurring if it has a one in two chance of naturally
>occurring and you take two shots at it? The easiest way to calculate that
>question is by asking what's the probability of it NOT occurring. The
>binomial theorem can be written in condensed form as only its coefficients:
>
>(a^2 + 2ab + b^b) : 1 2 1
>
>In this case, it's the far right "1" that is of interest, the chance that
>event didn't occur. And in this case, it's (1/2)^2, or 1/4th, or a 3/4
>probability that the event did occur at least once.
>
>Similarly, what's the probability that a one in three chance occurs if you
>(or God) take three shots at it.
>
>1 3 3 1
>
>Again, the probability becomes 1 - (1/3)^3, or 8/9ths.
>
>And what then is the probability if the chance of the truly cool event
>occurring is only 1 in a billion, but you get to take one billion shots at it
>(which is not a large number of shots in either population genetics or
>cosmology)? Again, the probability of it occurring at least one time in those
>billion shots is 1 - (1/1 billion)^1 billion, or .999999999.... out to 1
>billion decimal places, which is guaranteed certainty, or at least certainty
>close enough for government work.
>
>So long as you have a "barb of selection" (or a reified Maxwellian Daemon) in
>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 does
not occur in n trials is then (1 - 1/n)^n and the probability that it occurs at
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%.   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.

Ted
--
Ted Ashton ([log in to unmask]) | From the Tom Swifty collection:
Southern Adventist University    | "The doctors had to remove a bone from my
Deep thought to be found at      | arm", said Tom humorlessly.
http://www.southern.edu/~ashted  |

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