The realistic definition of transition probability in physics is well defined and constrains the probability to rational numbers. The abstract definition of probability in mathematics is also well defined, but it allows probability to be an arbitrary real number element of [0,1], whether rational or irrational. The difference is enormous and the conclusions diverge widely. The question is who do we track and when?
: We can provide two specific published examples among many others,
a-Numerical resolution of the 3D PDE of heat diffusion as a function of time in its most general case using the physical definition of probability.
b-Solve the statistical numerical integration for an arbitrary number of free nodes using the physical definition of probability. The trapezoidal ruler and the FDM-based sympson ruler would be just a special case.
Yes indeed. There is a deep difference in concept.
Classical probability is related to the number of favorable events you find among a certain number of independent events of equal likelyhood.
(In any probability book). However there are near smooth distributions also, not only discrete, For example a certain distribution for the heights of persons.
In quantum mechanics it is based on the wave function as Phi* Phi,
which gives a probability density. An integral over probability density give a probability to locate an electron in the volume or interval indicated by the integral.(In any quantum book)In particular the integral over all available space should be normlized to 1.
But there is no real difference as to what kind of number the answer gives
Classical or quantum probability may both give real numbers, in the
interval [0,1] Classical probability can also be dealt with using continuous distributions. The real number axis includes both rational or irrational numbers.