When would I use theoretical probability and when would I use experimental probability?

Question

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?

Answer 1

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.

Answer 2

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?