[extropy-chat] Coin Flip Paradox (was Randomness)

gts gts_2000 at yahoo.com
Mon Jan 29 01:46:56 UTC 2007

On Sun, 28 Jan 2007 18:05:39 -0500, The Avantguardian  
<avantguardian2020 at yahoo.com> wrote:

> I have been meaning to return to this particular issue
> because I am completely disattisified with the Axiom
> of Randomness as so stated.

It is, as you mention, a *frequentist* idea of randomness.

You note correctly that the frequentist interpretation of randomness (and  
of probability too, I would add) gives no account of the probability of  
*single* events, such as a single flip of a coin. This aspect of the  
frequency theory has always been known, and is the primary reason people  
like Karl Popper developed propensity theories of probability. It is also  
one reason to use Bayesian methods.

Both Baysianists and propensity theorists can talk meaningfully about the  
probability of single or rare events. Frequentists cannot.

> So contrary to intuition, the more
> times you flip the coin, the *less* likely you are to
> measure a frequency of heads equal to the *known*
> probability of getting getting heads on a coin flip in
> the first place. That in a nutshell is the Coin Flip
> Paradox and why I am a Bayesian.

I think your paradox here is based on either a false intuition or a  
misconception of frequentism. Frequentists do not argue that the observed  
frequency of heads should ever be *exactly* 0.5 in any finite number of  
observations. They argue merely that the observed frequency will  
*converge* on 0.5 as n tends to infinity.

> If probability were a limiting frequency on
> the other hand then it would be mathematically
> *guanteed* that the frequency would be equal to the
> probability at infinite n.

But that is exactly the case according to the Von Mises's first empirical  
law of probability. Probability is calculated as the limiting frequency  
(calculated as a proportion or as a percentage) as n goes to infinity. It  
can be shown by the way that this convergence of the observed frequency  
follows an inverse square rule, not unlike some other supposed empirical  
laws of the universe.


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