A probability space is a space Ω of "outcomes" along with a probability function P on some of Ω's subsets, such that P(Ω) = 1. (which is to say that Ε P(O) = 1, O in Ω.)

The probability of a disjoint countable union is the sum of the probabilities (in particular P(complement(A)) = 1 - P(A) where A is any subset of Ω.)

See also: set algebra (which is actually in English.)

Of course, ariels' definition, below, and my simpler one above (well, actually ariels' simpler one above, if the truth be known, conveyed to me in E2 /msg's - a technical detail, don't worry about it) while excellent for modeling the behaviour of 'random' events, do not effectively provide a definition of randomness itself.

To see this, consider the following probability space:

Let our Ω be {2+2=4,2+2=5,2+2=...}
let P be 1 where 2+2=4, 0 otherwise.
Obviously, it is not random that 2+2=4, therefore our 'probability space' has failed to distinguish 'random' from 'deterministic' events.

In ariels' fuller definition, below, we can easily imagine that the X is the set of outputs for some complex computer program, designed to mimic what's modelled by a given probability function, P. Again, we can see that, though the actual behaviour of the program may be excellently modelled by the probability space, as the output of a program is never random the probability space fails to distinguish random from nonrandom events. In the end, you have to send someone from the physics department to get your randomness from somewhere out in the quantum world.

So we can say that, while probability spaces are great for describing and modelling randomness, randomness itself is a prior notion, which we need in order to satisfy our intuitive understanding of what it is that a probability space is modelling. Hence the utility of a fair die in the definition below (or of the Bagatelle in the set algebra writeup.) Randomness is a lot trickier than we normally take it to be.


This node has an amusing history. It all started with a casual remark I dropped in the chatterbox, which ariels took issue with. In the ensuing discussion via /msg, I got him to define probability space, and I cheekily pasted his definition into a writeup for him and noded it (the mathematical part, above.) The /msg'd discussion ballooned out, and as a result ariels posted his refined and improved definition, which you see below. Having seen this, I decided to add the more philosophical part to my writeup, so it wouldn't be entirely useless. When I pressed submit, I was amazed to find that my writeup had been C!ed, more or less simultaneously with (but slightly before) my altering it. It transpired that my writeup had been mis-C!ed, the intention of the C!-er being to C! ariels' definition, rather than mine - understandable, as mine was a bare 5 lines of mathematical gobbledygook, whereas ariels has 15 lines of this, and it's better gobbledygook.

Now ask yourself, is this synchronicity a mere random event, or does it argue some implicate order in the nodeflow? Anyway, that's why I'm C!ing ariels' writeup, below.

A probability space on a set X is a measure space (which see) (X,B,P) for which P(X)=1.

One oddity of probability theory is that all objects of Analysis have different names. In this case, the measure P is called a probability, and the sets A∈B (usually known as measurable sets) are called the events in the probability space. These are the sets for which a probability is defined.

Examples

  1. X={1,2,3,4,5,6}, B is all 64 subsets of X, and P(A)=|A|/6 (the probability of a set is its size divided by 6).

    We can verify (X,B,P) is a probability space. It is commonly used to describe a single roll of a fair die. The event "the die comes up even" is the set {2,4,6}∈B; its probability is 1/2.

  2. X=R (the real numbers), B = the σ algebra of Lebesgue sets (a technical detail; apart from noting not every subset of R is in B, just ignore it), and
    P(A) = ∫-∞+∞ 1A(x) 1/(2π)1/2 exp(-x2/2) dx,
    where the characteristic function of A is 1A(x)=1 for x∈A and =0 otherwise.

    This describes a random variable which follows the normal distribution.

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