Circular Definition of Experiment in probability

Question

I was trying to understand what an experiment was in the theory of probability. I found several definitions.

Definition by Wikipedia

Any procedure that can be infinitely repeated and whose outcomes are well-defined.

Standard Definition

An experiment is a probability space $(\Omega, \Sigma, \mathbb{P})$

Issue with the definitions

So my understanding is that an experiment is used to define what a sample space is and what its outcomes are. From this we can define events and the event space. From these we can define a probability measure and therefore define the triplet $(\Omega, \Sigma, \mathbb{P})$ to be a probability space.

Therefore a probability space is defined starting from an experiment. But an experiment is defined starting from a probability space. This is a circular definition!

Possible solution to the issue

My guess is that the correct definition of an experiment is

Any procedure that can be infinitely repeated and whose outcomes are well-defined.

Or maybe the one given by Grimmett & Welsh:

Any procedure whose consequence is not predetermined.

But surely not the one with the probability space. Rather, I would say that an experiment is represented by a probability space, but not defined from it!

Is this correct? Or do we allow circular definitions?

Answer 1

So my understanding is that an experiment is used to define what a sample space is and what its outcomes are.

What you hope for them to be, it depends on the design.

See "experimental method" and "experimental design".

From this we can define events and the event space. From these we can define a probability measure and therefore define the triplet (Ω,Σ,P) to be a probability space. Therefore a probability space is defined starting from an experiment.

OK.

But an experiment is defined starting from a probability space. This is a circular definition!

Chicken and the egg. Figure out one, probably the experiment and it's design, then decide what to measure.

You could decide what you enjoy measuring and performing statistical analysis on the most, then create an experiment.

I was trying to understand what an experiment was in the theory of probability.

Here's one example:

In "An Introduction to Random Sets" by Hung T. Nguyen on page 13, in section "3 - Generalities on Probability", subsection "1.2 Mathematical Models for Random Phenomena", Definition 1.1:

A mathematical model for a random experiment is a probability space ($\mathcal{\Omega, A, P})$, where:

$\quad\alpha$) $\Omega$ is a set, representing the sample space of the experiment,

$\quad\beta$) $\mathcal{A}$ is a σ-field (representing events), i.e.:

$\qquad\,\,\,\, i)\, \Omega \in \mathcal{A}$

$\qquad\; ii)\, I\!\mathcal{f \, A \in A, \, then \, A^c \in A}$

$\qquad iii)\, I\!\mathcal{f \, A_n \in A \, for \, n \ge 1, \, then \, \underset{n \ge 1}\bigcup A_n \in A.}$

The pair ($\Omega, \mathcal{A}$) is called a measurable space.

$\quad\gamma$) $\mathcal{P : A} \rightarrow [0, 1]$ is a probability measure, i.e.:

$\qquad\,\,\,\, a) \; \mathcal{P}(\Omega) = 1$

$\qquad\,\,\,\, b) \;I\!\mathcal{f} \, \{A_n, n \ge 1\}$ is a sequence (finite or infinitely countable) of pairwise disjoint (i.e., $A_n \cap A_m = ∅ \, for \, n \ne m$) elements of $\mathcal{A}$, then $\mathcal{P}(\underset{n \ge 1}\bigcup A_n) = \underset{n \ge 1} \sum P(A_n)$, a property that is referred to as σ-additivity of $\mathcal{P}$.

$\,$ For example, in sampling from a finite population $U$ with $\#(U) = N$, if we decide to select samples of given size $n$, then $\Omega = 2^U\!$, $\mathcal{A}$ = power set of 2$^U\!$, and $\mathcal{P : A} \rightarrow [0, 1]$ is defined as $\mathcal{P}(\mathbb{A}) = \underset{A \in \mathbb{A}} \sum P(A), \mathbb{A} \subseteq 2^U$, where for $A \in 2^U$,

Answer 2

In a rigorous treatment of probability, the word "experiment" is not even used. You just define a sample space and probability measure without mentioning any "experiment". Now, you may be thinking in your mind that each element of the sample space corresponds to one possible outcome of a real-world "experiment" like rolling a die, but there is no need to give a precise definition to the word "experiment" (and also I don't know how to give this term a precise definition even if we wanted to).

Answer 3

But surely not the one with the probability space. Rather, I would say that an experiment is represented by a probability space, but not defined from it!

When writing about mathematical models of real-world phenomena, it is common to speak about a part of the model with the word for the real concept it models.

For example, someone who is trying to model parliamentary negotiations, might say something like: "Let a party be a function that assigns to each issue space a weighted preference ordering ..." even though they know perfectly well that a political party in reality is something different than the function in his model.

The definition you have found in Wikipedia sounds like it's trying to describe a class of real-world phenomena, as part of an explanation of how to apply probability to real-world situations.

What you call a "standard definition" is part of a description of the mathematical model. (Like littleO, I am skeptical how standard that actually is, but that's not really the point, I think).

The two definitions use the same word for things at two different levels of abstraction. This is normal and usually not a cause of confusion once you're aware that they are two different things.