[extropy-chat] what is probability?

[gts]

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