Random Experiment

probability-theory

Definition

Random Experiment

A random experiment is a well-specified procedure that can be repeated under comparable conditions and produces exactly one realised outcome from a known sample space $\Omega$, while the result of a particular performance is uncertain beforehand:

$$\mathcal{E}⟶\omega ,\omega \in \Omega .$$

One performance of the experiment is called a trial.

Requirements

• The procedure and observation rule are specified clearly enough to determine what counts as an outcome.
• The set of possible outcomes is known before a trial.
• The realised outcome is not known with certainty before the trial.
• Repetitions are performed under conditions treated as comparable by the probability model.

Tossing a coin

Tossing a coin and recording its upper face is a random experiment with

$$\Omega =\{H,T\}.$$

Each toss is a trial and produces one realised outcome, $H$ or $T$. The experiment specifies the possible outcomes, but not which one will occur on a given toss.

Warning

Randomness does not imply that all possible outcomes are equally likely. It states only that the realised outcome is uncertain within the chosen model. This uncertainty may arise from an intrinsically random mechanism or from relevant conditions that the model does not track.