Probability Space - Introduction

Introduction

A probability space is a mathematical triplet ( Ω, P) that presents a model for a particular class of real-world situations. As with other models, its author ultimately defines which elements Ω, and P will contain.

• The sample space Ω is a set of outcomes. An 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. What differences matter depends, of course, on the kind of analysis we want to do. This leads to different choices of sample space.

• The σ-algebra is a collection of all and only events (not necessarily elementary) we would like to consider. Here, an "event" is a set of zero or more outcomes, i.e., a subset of the sample space. An event is considered to have "happened" when the outcome is a member 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 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 of "7 pips" and "odd number of pips" have also happened.

• The probability measure P is a function returning an event's probability. A probability is a real number between zero (impossible events have probability zero, though probability-zero events need not be impossible) and one (the event happens almost surely). Thus P is a function . The probability measure function must satisfy a simple requirement: the probability of a union of two (or countably many) disjoint events must be equal to the sum of probabilities of each of these events. For example, if two events are Heads and Tails, then the probability of Heads-or-Tails must be equal to the sum of probabilities for Heads and Tails).

Not every subset of the sample space Ω must necessarily be considered an event: some of the subsets are simply not of interest, others cannot be “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 "irrational numbers between 60 and 65 meters"