Probability space

is a mathematical construct that provides a formal model of a random process or "experiment".

For example, one can define a probability space which models the throwing of a die.

In the example of the throw of a standard die, When an experiment is conducted, it results in exactly one outcome

must be so defined that if the experiment were repeated arbitrarily many times, the number of occurrences of each event as a fraction of the total number of experiments, will most likely tend towards the probability assigned to that event.

In modern probability theory, there are alternative approaches for axiomatization, such as the algebra of random variables.

must necessarily be considered an event: some of the subsets are simply not of interest, others cannot be "measured".

In a different example, one could consider javelin throw lengths, where the events typically are intervals like "between 60 and 65 meters" and unions of such intervals, but not sets like the "irrational numbers between 60 and 65 meters".

consisting of: Discrete probability theory needs only at most countable sample spaces

The probability measure takes the simple form The greatest σ-algebra

If this sum is equal to 1 then all other points can safely be excluded from the sample space, returning us to the discrete case.

A formulation stronger than summation, measure theory is applicable.

Initially the probabilities are ascribed to some "generator" sets (see the examples).

If the experiment consists of just one flip of a fair coin, then the outcome is either heads or tails:

Thus her incomplete information is described by the partition Ω = A1 ⊔ A2 = {HHH, HHT, THH, THT} ⊔ {HTH, HTT, TTH, TTT}, where ⊔ is the disjoint union, and the corresponding σ-algebra

His partition contains four parts: Ω = B0 ⊔ B1 ⊔ B2 ⊔ B3 = {HHH} ⊔ {HHT, HTH, THH} ⊔ {TTH, THT, HTT} ⊔ {TTT}; accordingly, his σ-algebra

We assume that sampling without replacement is used: only sequences of 100 different voters are allowed.

We also take for granted that each potential voter knows exactly his/her future choice, that is he/she does not choose randomly.

Bryan knows the exact number of voters who are going to vote for Schwarzenegger.

His incomplete information is described by the corresponding partition Ω = B0 ⊔ B1 ⊔ ⋯ ⊔ B100 and the σ-algebra

is the σ-algebra of Borel sets on Ω, and P is the Lebesgue measure on [0,1].

In this case, the open intervals of the form (a,b), where 0 < a < b < 1, could be taken as the generator sets.

Each such set can be ascribed the probability of P((a,b)) = (b − a), which generates the Lebesgue measure on [0,1], and the Borel σ-algebra on Ω.

Here one can take Ω = {0,1}∞, the set of all infinite sequences of numbers 0 and 1.

These two non-atomic examples are closely related: a sequence (x1, x2, ...) ∈ {0,1}∞ leads to the number 2−1x1 + 2−2x2 + ⋯ ∈ [0,1].

In fact, all non-pathological non-atomic probability spaces are the same in this sense.

Basic applications of probability spaces are insensitive to standardness.

However, non-discrete conditioning is easy and natural on standard probability spaces, otherwise it becomes obscure.

is too "large", i.e. there will often be sets to which it will be impossible to assign a unique measure.

, for example the Borel algebra of Ω, which is the smallest σ-algebra that makes all open sets measurable.

This extends to a (finite or countably infinite) sequence of events.