Definition:Experiment

Definition

An experiment (or trial) is defined as:

a course of action whose consequence is not predetermined.

An experiment $\mathcal E$ can be formulated mathematically by means of a probability space, which consists of:

• The sample space $\Omega$: that is, the set of all possible outcomes of the experiment;

• The event space $\Sigma$: that is, the list of all the events which may occur as the consequences of the experiment;

• The probability measure $\Pr$ on the event space: that is, the likelihood of the happening of each of the events in the event space.

With this definition, $\mathcal E$ is a measure space $\left({\Omega, \Sigma, \Pr}\right)$ such that $\Pr \left({\Omega}\right) = 1$.

Example

Let $\mathcal E$ be the experiment of throwing a standard 6-sided die, to see whether the number thrown is greater than $4$.

• The sample space of $\mathcal E$ is $\Omega = \left\{{1, 2, 3, 4, 5, 6}\right\}$.

• The event space of $\mathcal E$ is: $\Sigma = \left\{{\forall \omega \in \Omega: \omega \le 4, \omega > 4}\right\}$.

• The probability measure is defined as: $\displaystyle \forall \omega \in \Omega: \Pr \left({\omega}\right) = \frac 1 6$.

Sources

• Geoffrey Grimmett: Probability: An Introduction (1986): $\S 1.1, \ \S 1.2$