[extropy-chat] what is probability?
gts
gts_2000 at yahoo.com
Wed Jan 17 17:08:36 UTC 2007
On Wed, 17 Jan 2007 11:34:51 -0500, Rafal Smigrodzki
<rafal.smigrodzki at gmail.com> wrote:
> ### Now, is that true? Let's say you have an unfair coin, which 99% of
> the time drops heads and only 1 % tails. Would you call the results
> of tossing it a "completely random sequence"?
Yes, because the tosses are independent and no gambling system could be
devised that would result in a greater than 99% win rate.
> What if the coin is only slightly unfair, dropping 50.0001 % heads? Is
> the resulting sequence
> random?
Yes. Same argument.
> Note that any deviation from the principle of indifference allows a
> player who knows about it to make arbitrarily large amounts of money
> betting on the outcomes
But that is not what the probability theorists mean by "gambling system".
From
http://www.philosophyprofessor.com/philosophies/impossibility-of-a-gambling-system-principle.php:
"The key condition is that the limiting frequency of the characteristic
concerned should be the same for all partial sequences we could select
from the collective, provided only that whether a given term in the
collective is taken into a partial sequence is independent of whether that
term manifests the characteristic concerned."
Here is how that sentence translates to a simple unfair coin-flip example
in which heads is highly favored:
"The key condition for satisfying the condition of randomness is that the
% heads is the same for all sub-sequences selected as favorable by the
supposed betting system as it is for the entire sequence, provided that
the gambling system doesn't include the unfair luxury of knowing in
advance which sub-sequences contain a greater than average % of heads."
You aren't using a gambling system when you're just playing the averages.
Gambling systems are attempts to *beat* the averages.
Make sense?
-gts
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