Does a "pure" real valued probability make any sen

From: SkanderH (skander_at_hrt.dis.titech.ac-dot-jp.no-spam.invalid)
Date: 12/01/04 

Date: 30 Nov 2004 22:28:31 -0600

According to most text books the definition of the probabilty of an
outcome is that if you could (ideally) perform an experiment to test
this outcome a (infinite) large number of times then the probability
of the outcome is the ratio of the number of successful outcomes to
the total number of experiments.
This suggests two things:

* that a probabilty function belongs to the [0,1] interval, which is a
well known principle stated and used throught scientific literature.

* that a probability function with pure real values (non rational)
such as sqrt(3)/2 or pi/4 does not make any sense, since such values
do not represent a ratio of two integers and thus have no meaning
from the point of view of the above stated "operational" definition
of probability.

I have very little training in advanced mathematics, and I dont know
if this idea is worth anything.
But here are my questions:

1 - does a pure real valued probability have any meaning? if yes,
what?

2 - if no, that is if, probability functions should only have rational
values in the [0,1] interval, then what are the implications for
theories that rely heavly on probability such as QM or non
deterministic signal processing?

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