I'm reading a book on probability theory and they say that an experiment is

Any procedure that has not got a pre-determined outcome

In the Wikipedia page they, instead, that an experiment should:

  • be infinitely repeatable
  • have well-defined set of possible outcomes

Now, I really don't see why it has to be infinitely repeatable. I mean, it makes sense only for frequentist probability right? Surely not for bayesian one.

Anyway, my question is: then what is NOT an experiment? what are some examples of procedures that are not experiments?

My Solution

My idea is that the following are experiments:

  • Tossing a fair coin, and looking at which side is facing upward after landing.
  • Tossing a fair dice, and looking at which side is facing upward after landing.
  • Picking up a numbered ball from within a box (where balls are placed randomly and the picking up mechanism is fair) and looking at the number of the ball.

However, I can't quite make up any sensible example of something that is NOT an experiment. For instance, I think that also the following is an experiment:

  • Tossing a non-fair coin that always lands on HEAD and looking at the side facing upwards.

because even though the coin is biased, it is infinitely repeatable and has well-defined set of possible outcomes.

What are some examples of non-experiments?

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    $\begingroup$ I wouldn't only read these things in the context of probability theory. Say you melt a lump of copper and measure the temperature that it takes to do that. You draw a conclusion about the melting point of copper, and you expect your conclusion to hold in other instances of melting copper. There you have "infinite repeatability." $\endgroup$ – Michael Hardy Aug 14 '18 at 12:51

An example of a procedure that is not infintely repeatable is drawing a marble from an urn with a finite number of red and blue marbles in it, without replacement, and observing the color.

It has a well-defined set of possible outcomes (i.e. $\{\text{Blue},\text{Red}\}$), but it is not infinitely repeatable. Eventually we will run out of marbles.

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    $\begingroup$ You can fill the urn back and restart from the beginning. I am not sure that is what is meant by 'infinitely repeatable". $\endgroup$ – nicomezi Aug 14 '18 at 12:27
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    $\begingroup$ In an experiment, the probability space $(\Omega,\mathcal{F},\mathbb{P})$ should stay constant. However, when we are repeatedly taking marbles out of the urn, the probabilities of picking a red or blue one change constantly, so it is not a valid experiment. If you did replace the marble, then it would be a valid experiment. $\endgroup$ – molarmass Aug 14 '18 at 12:33
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    $\begingroup$ That basically assures that the outcome space $\Omega$ is unambiguously defined, i.e. we all agree what could possibly happen. That is necessary to construct $\mathcal{F}$ and $\mathbb{P}$. $\endgroup$ – molarmass Aug 14 '18 at 13:00
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    $\begingroup$ For instance, the outcome space $\Omega=\{x:x \text{ is one of the 5 best songs of 2018}\}$ is not well-defined because (1) we do not necessarily agree on what the best songs are, and (2) 2018 is not over yet, so there may be better songs in the future. This is not a suitable outcome space. $\endgroup$ – molarmass Aug 14 '18 at 13:03
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    $\begingroup$ What can be an experiment is the following: draw $k$ marbles out of an urn with $n$ marbles without replacement, with $k<n$. Then put all marbles back and repeat. The difference that in this case, drawing $k$ marbles is just 1 trial, and each trial starts with the same initial conditions. So the probabilities stay constant. In my previous example, every marble we drew from the urn was a separate trial, which is a subtle but crucial difference. $\endgroup$ – molarmass Aug 14 '18 at 13:51

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