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For a coin toss, the sample space contains "Heads" and "Tails" for the possible outcomes and I can figure out that the probability of "Heads" is 0.5 if it is a fair coin. The probability, 0.5, is the theoretical probability as long as we know all the possible cases for a coin toss. Therefore, without throwing a coin a thousand times, I just know the the probability of "Heads" will occur is 0.5 and we don't need to experiment to find out the probability of "Heads" to happen. But when would I need to have experimental probabilities?

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    $\begingroup$ When you have a sequence of coin tosses, but don't know the probability of "Heads". $\endgroup$
    – Sergio
    Sep 6 '20 at 10:10
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You would almost never "know" the theoretical probability. The exception are maybe the most simple examples like chance of drawing a winning vote in a lottery, or chance of drawing particular hand in card game, etc. Even for the textbook example of a coin, the empirical probabilities are not exactly 1/2, because you are usually not tossing the ideal coin in a vacuum, using a robotic arm that has 100% reproducibility of it's movements. Usually the probabilities depend on many different factors and it is not possible to account for them all to figure out the probability exactly, here we use experiments to learn the probabilities from the data. As for "known", or "theoretical" probabilities, they occur either in probability textbooks, or are just guesses to be used as a priors, or for making back-of-the-envelope guesstimates, but rather nowhere else.

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Before to give a proper answer to your question we have to give an answer to another one: what are probabilities ?

Reply to this question is far from trivial. Exist several definition of probability; for a short review read here:https://en.wikipedia.org/wiki/Probability#Interpretations. Any reply to your question should be preceded from a disentanglement about the definition.

"Experimental probability" sound like frequentist probability. The idea behind this approach starting from "ignorance" about the quantification of probabilities of some events. Frequentist approach refuse the priori of equiprobability assumption. Staying at coin tossing example: $P(head)=0,5$ and $P(tail)=0,5$ is refused as priori knowledge.

Following a frequentist approach: after the experiment, under replicability condition and when the proper convergence are at acceptable level, we achieve a good estimate of the two probabilities.

In theoretical examples we can always consider probabilities are given. No experiments, only calculus follow.

In practical cases, priori about probabilities, in terms of equiprobability or others assumption, remain always debatable. Unfortunately even the reliability of conditions like: replicability and proper convergence are always debatable. In case like gambling the equiprobability hypothesis is widely accepted, in most case tacitly. Sometimes experiment are conducted in critique or confirmation fashion. In many others case frequentist approach are follow. Ultimately, what to do in any practical case is a your choice.

Debate about the meaning of probability have a long and interesting history. Only one warning. Even if in science the frequentist approach is far from discard, several his drawback was discovered. Subjective approach become always more relevant. This discussion is strongly related: Is independence subjective?

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  • $\begingroup$ Do we still consider the sample space for experimental probabilities as it is considered for the theoretical probability? My question might not make sense to you, please understand I have just started to think about this topic. $\endgroup$
    – StoryMay
    Sep 7 '20 at 12:14
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    $\begingroup$ Yes. At least in this context a well conducted experiments suppose that you are aware about all the possible outcomes. The sample space and all possible events (sigma algebra) are the same regardless the meaning of probability you deal with. $\endgroup$
    – markowitz
    Sep 7 '20 at 12:55
  • $\begingroup$ I wonder what I should do If I want to know probabilities for students' favourite breakfast among "Cereal", "Eggs", "Toast", "Orange Juice". What could be the sample space and theoretical probabilities for each favourite breakfast? Do I need an experiment for probabilities, for example, I survey 20 students and draw a probability distribution? $\endgroup$
    – StoryMay
    Sep 7 '20 at 13:26
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    $\begingroup$ The sample space can be the list that you gave. Surveys and experiments are different things. Maybe survey is what you need here. However you have to define carefully the events that you are interested in. $\endgroup$
    – markowitz
    Sep 7 '20 at 19:45
  • $\begingroup$ If the sample space consists of the list of options, then can I say the theoretical probabilities are 1/4 for each option? Although the theoretical probability is 1/4 for each option, to me it feels like that's what we expect but we don't know the true theoretical probability for each option, that's why we conduct a survey or an experiment in real life to estimate the true values from the sample. Hope I am not confusing you. $\endgroup$
    – StoryMay
    Sep 8 '20 at 10:07

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