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?
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
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
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.