Probability space, Sample space, Event, and Elementary event

1. Overview

Probability space or a probability triple $(\Omega ,F,P)$ is a mathematical construct that models a real-world process (or “experiment”) consisting of states that occur randomly.

Sample space (also called sample description space or possibility space) of an experiment or random trial is the set of all possible outcomes or results of that experiment.

Event is a set of outcomes of an experiment (a subset of the sample space) to which a probability is assigned.

Elementary event (also called an atomic event or sample point) is an event which contains only a single outcome in the sample space.

2. Description

2.1 Probability Space

A probability space is constructed with a specific kind of situation or experiment in mind. One proposes that each time a situation of that kind arises, the set of possible outcomes is the same and the probabilities are also the same.

A probability space consists of three parts:

• $\Omega $: A sample space which is the set of all possible outcomes.
• $F$: A set of events where each event is a set containing zero or more outcomes.
• $P$: The assignment of probabilities to the events; that is, a function P from events to probabilities.

2.2 Sample space

A sample space is usually denoted using set notation, and the possible ordered outcomes are listed as elements in the set. It is common to refer to a sample space by the labels S, Ω, or U (for "universal set"). The elements of a sample space may be numbers, words, letters, or symbols. They can also be finite, countably infinite, or uncountably infinite.

A set $\Omega $ with outcomes $s_{1}, s_{2},\cdots , s_{n}$ (i.e. $\Omega =\left \{ s_{1}, s_{2},\cdots , s_{n} \right \}$) must meet some conditions in order to be a sample space:

2.2.1 mutually exclusive

If $s_{j}$ takes place, then no other $s_{i}$ will take place,
$\forall i,j=1,2,\cdots ,n \: i \neq j$

2.2.2 collectively exhaustive

On every experiment (or random trial) there will always take place some outcome $s_{i}\in \Omega \:  for \: i\in {1,2,\cdots ,n}$

2.2.3 right granularity

The sample space ($\Omega$) must have the right granularity depending on what we are interested in. We must remove irrelevant information from the sample space. In other words, we must choose the right abstraction (forget some irrelevant information).

For instance, in the trial of tossing a coin, we could have as a sample space $\Omega _{1}=\{H,T\}$, where H stands for heads and  T for tails. Another possible sample space could be $\Omega _{2}=\{H\&R,H\&NR,T\&R,T\&NR\}$. Here R stands for rains and NR not rains. Obviously, $\Omega _{1}$ is a better choice than $\Omega _{2}$ as we do not care about how the weather affects the tossing of a coin.

2.3 Event

A single outcome may be an element of many different events, and different events in an experiment are usually not equally likely, since they may include very different groups of outcomes. For example, An even number from throwing a die one or two times.

2.4 Elementary event

Using set theory terminology, an elementary event is a singleton. Elementary events and their corresponding outcomes are often written interchangeably for simplicity, as such an event corresponds to precisely one outcome. For example, A Head from a flip of a fair coin.

The following are examples of elementary events:

• All sets {k}, where k ∈ N if objects are being counted and the sample space is S = {0, 1, 2, 3, ...} (the natural numbers).
• {HH}, {HT}, {TH} and {TT} if a coin is tossed twice. S = {HH, HT, TH, TT}. H stands for heads and T for tails.
• All sets {x}, where x is a real number. Here X is a random variable with a normal distribution and S = (−∞, +∞). This example shows that, because the probability of each elementary event is zero, the probabilities assigned to elementary events do not determine a continuous probability distribution.

3. Reference

https://en.wikipedia.org/wiki/Probability_space

https://en.wikipedia.org/wiki/Sample_space

https://en.wikipedia.org/wiki/Elementary_event

https://www.mathsisfun.com/data/probability.html

https://en.wikipedia.org/wiki/Event_(probability_theory)