The classical approach

Due to our ignorance about the outcome of, say, a cast die, and because there is no indication for us to think one outcome more likely than any other, we must give them all an equal probability.

The probability of an event [math]A[/math] (written as [math]P(A)[/math] for example, casting an even number with a die) is defined as the number of favorable outcomes ([math]n[/math], or 3 – the numbers on a dice that are even) divided by the number of possible cases ([math]m[/math], or in this example 6, the number of sides of the die):

[display]P(A) = \frac{n}{m}[/display]

(therefore in the example 3/6, or a half). From this definition (almost) the whole of probability theory follows. This definition is not unproblematic, as it seems to rest on circular reasoning: probability is defined by dividing into equally probable cases, but how can we know they really are equally probable if we’re looking for a definition of probability in the first place?

Pierre-Simon Laplace (from Wikimedia Commons)

Conceptually, what does it mean that an event has a certain probability? By the time most of the fundamentals of probability theory had been worked out, Newtonian science and enlightenment philosophy coloured what was thought about probability. There were no longer gods that ruled the universe on a whim. Reality was understood as a giant clockwork mechanism, where natural laws were followed precisely and for all eternity. This deterministic view influenced the first philosophical interpretations of what we mean by probability: probability is a consequence of our ignorance about the world. In 1814, the mathematician Laplace imagined Laplace1814 a very powerful being who knew the precise location of every single particle in the world, and could compute, using the laws of physics, their trajectories and future positions. Now, imagine a die is cast. The being will be able to compute the precise positions of all the particles in the die afterwards and therefore know the outcome in advance. For this being, which philosophers fondly call “Laplace's demon”, probability as a concept would be useless. Laplace therefore interpreted probability as a measure of our ignorance.

These days the idea that probability is solely a measure of our ignorance is not very widely held, although several modern interpretations of probability take at least a similar approach and see probability as a degree of belief, and therefore also firmly set it on the "epistemological" side of the two faces of probability.