Following up on this question: Definition of an "Experiment" in Probability

According to Wikipedia, an experiment is a procedure that can be infinitely repeated and has a set of possible outcomes. Why should we care if the experiment can be infinitely repeated?

An experiment is modeled as a probability space with each outcome being a point in the sample space. There's nothing in the definition of a probability space that indicates that the experiment should be infinitely repeatable. Even an experiment that hypothetically can only be repeated a finite number of times could be correctly modeled as a probability space, regardless of whether its sample space is finite or not.

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    $\begingroup$ It is a frequentist view. The probability of an event is the fraction of the time it occurs in an infinite sequence of trials. This is contrasted with the Bayesian view, but that is more subtle both philosophically and mathematically. So most introductory presentations use the frequentist view. $\endgroup$ – Ian Feb 6 '18 at 18:53
  • $\begingroup$ @Ian: I see. But that's just another way of defining probability via computation. Crudely speaking that's a physically intuitive way of defining it. The way it's typically defined in grad-level probability is just assigning a number to a measurable set. Do we need the infinitely repeatable requirement even in the latter case? $\endgroup$ – Shirish Kulhari Feb 6 '18 at 18:58
  • $\begingroup$ @Ian: Also, wouldn't the frequentist way of defining probability break down if the sample space is uncountable? $\endgroup$ – Shirish Kulhari Feb 6 '18 at 19:00
  • $\begingroup$ It is not obvious that the measure theory definition means anything; this frequentist view is one way to give it meaning. And no, when the sample space is uncountable nothing breaks down except that typically there are many events with probability zero. $\endgroup$ – Ian Feb 6 '18 at 19:08

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