Probability Disambiguation

Probability is a meta-physical concept.

This is the reason why it is difficult for many people to fully grasp and work with. We are accustomed to interacting with concepts that are or can be connected with the physical world that we experience through our senses. We understand things like location and size, because we experience those physical concepts through our senses.

Probability, chance, is not something that is experienced through the senses, and we try to form physical analogues in hopes that these will explain it to those of us with a weaker imagination (apologies for the arrogance; it's meant to motivate, not to insult).

Probability is a meta-physical concept.

Let's take, for an example, a particle, suspended in space, at a point, A, and that point, A, happens to be nearby your face. The familiar physical interpretation of your surrounding universe, in that case, is that there exists a particle, in front of your face, at point, A.

The probabilistic interpretation of the universe surrounding you, in that case, is that there exists a particle, and it exists everywhere, not just at point, A, but with a varying density of existence. It's existence is very sparse at most loci in the universe, but is very dense at point, A; it still exists everywhere, however.

The gap between these two interpretations is the reason why the idea that an electron, existing around a nucleus, has undetermined, stochastic location is mystical to the physicist, while the mathematical scientist shrugs his shoulders and nods a 'okay, go on' to the same.

The domain of existence of our particle was a spacial one. A stochastic variable might have a univariate Gaussian density, in which case, it's domain of existence would be the real field, a numeric domain of existence. In the same way as the particle, the stochastic variable has all real numeric values, but has varying density over the real field.

An analogue many-a-times used to embed the concept of a stochastic variable into the mind of the student is to view it as a collection of determined numbers, and its density as a sort of normalised population histogram. This is a convenient view to use, but it is not a faithful one. The inadequacy of this approach exposes itself in the fact that students have much trouble in more advanced applications of probability theory. The problem with this view is that it does not define a stochastic variable, it is a physical situation for which a stochastic variable may be used as a model. The stochastic variable itself is not a collection of numbers, it is not a number in the field. The stochastic variable is a mathematical meta-object.