Editing Probability space (section)

== Introduction ==
[[File:Dice measure.svg|thumb|400px|Probability space for throwing a die twice in succession: The sample space <math>\Omega</math> consists of all 36 possible outcomes; three different events (colored polygons) are shown, with their respective probabilities (assuming a [[discrete uniform distribution]]).]]
A probability space is a mathematical triplet <math>(\Omega, \mathcal{F}, P)</math> that presents a [[mathematical model|model]] for a particular class of real-world situations. As with other models, its author ultimately defines which elements <math>\Omega</math>, <math>\mathcal{F}</math>, and <math>P</math> will contain.
* The [[sample space]] <math>\Omega</math> is the set of all possible outcomes. An [[Outcome (probability)|outcome]] is the result of a single execution of the model. Outcomes may be states of nature, possibilities, experimental results and the like. Every instance of the real-world situation (or run of the experiment) must produce exactly one outcome. If outcomes of different runs of an experiment differ in any way that matters, they are distinct outcomes. Which differences matter depends on the kind of analysis we want to do. This leads to different choices of sample space.
* The [[σ-algebra]] <math>\mathcal{F}</math> is a collection of all the [[Event (probability theory)|event]]s we would like to consider. This collection may or may not include each of the [[Elementary event|elementary]] events. Here, an "event" is a set of zero or more outcomes; that is, a [[subset]] of the sample space. An event is considered to have "happened" during an experiment when the outcome of the latter is an element of the event. Since the same outcome may be a member of many events, it is possible for many events to have happened given a single outcome. For example, when the trial consists of throwing two dice, the set of all outcomes with a sum of 7 [[Pip (counting)|pips]] may constitute an event, whereas outcomes with an odd number of pips may constitute another event. If the outcome is the element of the elementary event of two pips on the first die and five on the second, then both of the events, "7 pips" and "odd number of pips", are said to have happened.
* The [[probability measure]] <math>P</math> is a [[set function]] returning an event's [[probability]]. A probability is a real number between zero (impossible events have probability zero, though probability-zero events are not necessarily impossible) and one (the event happens [[almost surely]], with almost total certainty). Thus <math>P</math> is a function <math>P : \mathcal{F} \to [0,1].</math> The probability measure function must satisfy two simple requirements: First, the probability of a [[Countable set|countable]] union of mutually exclusive events must be equal to the countable sum of the probabilities of each of these events. For example, the probability of the union of the mutually exclusive events <math>\text{Head}</math> and <math>\text{Tail}</math> in the random experiment of one coin toss, <math>P(\text{Head}\cup\text{Tail})</math>, is the sum of probability for <math>\text{Head}</math> and the probability for <math>\text{Tail}</math>, <math>P(\text{Head}) + P(\text{Tail})</math>. Second, the probability of the sample space <math>\Omega</math> must be equal to 1 (which accounts for the fact that, given an execution of the model, some outcome must occur). In the previous example the probability of the set of outcomes <math>P(\{\text{Head},\text{Tail}\})</math> must be equal to one, because it is entirely certain that the outcome will be either <math>\text{Head}</math> or <math>\text{Tail}</math> (the model neglects any other possibility) in a single coin toss.

Not every subset of the sample space <math>\Omega</math> must necessarily be considered an event: some of the subsets are simply not of interest, others cannot be [[Non-measurable set|"measured"]]. This is not so obvious in a case like a coin toss. 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".