I am not quite sure whether this belongs on math SE. Anyway, my question is this. In math, probability can theoretically be any real number between 0 and 1 inclusive. But what about in the real world? I don't think an event in the real world can have, say, pi percent probability. So, in the real world, is probability discrete, or continuous?
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I don’t think the ratio of a circle’s circumference to its diameter is pi in the real world either... no measurement is perfect.– Cotton Headed NinnymugginsJul 16, 2021 at 14:39
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Probability does not reside in the real world, only in our models of it, and reflects our state of knowledge in addition to external facts. An event cannot have probability π for the simple reason that π>1, but one can come up with a quantum system our model of which will assign probability 1/π to some outcome.– ConifoldJul 16, 2021 at 21:38
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@Conifold I did not mean pi, I meant pi percent, in other words, pi/100.– user107952Jul 16, 2021 at 23:36
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π/100 is just as good as 1/π or 1/2.– ConifoldJul 17, 2021 at 3:14
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It's worth noting that discrete and continuous are not the only possibilities. That being said, any probability measure can be expressed as a linear combination of a discrete measure, a singular continuous measure, and an absolutely continuous measure.– SandejoJul 17, 2021 at 16:46
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1 Answer
Probability is calculated as the ratio of the number of favorable outcomes out of the number of mutually exclusive possible outcomes (which I'll refer to as x/n here).
The answer to your question depends on whether the set of possible outcomes is countable and finite in the real world — which is not a settled issue.
If the number of possible outcomes is countable and finite, then a subset would be as well. Both x and n would be integers and the ratio x/n would therefore be rational. The set of rational numbers is discrete and not continuous, so then probabilities would necessarily be discrete too.
However, if the set of possible outcomes is infinite, even if it is countable, you can get irrational probabilities. For example, the density of all square-free integers (that is, the probability of choosing such an integer out of the infinite number of natural numbers) is 6 / (pi^2), an irrational number. You can research more about this on the Wikipedia entry on natural density.
And there's also a possibility that there are some aspects of nature that are continuous, which would also allow for continuous probabilities.
