So I understand that the probability a<x<b is the definite integral from a to b of tye probability density function and that makes sense. If we use that same definition to define the probability that x is equal to some value, then we would get 0. And yet, it doesn’t really make sense to say that probability of something is 0 if it’s still possible for it to happen. Furthermore, this introduces the paradox that adding up a bunch of zero probabilities somehow gives us a total property of 1. You can’t add up zero things and get a non zero result. That makes no sense. However, you can add up an arbitrarily large number of arbitrarily tiny pieces and and get a non zero real number result. This is precisely what integration does. So why don’t we just say that the probability of a single event in a continuous distribution is simply undefined? i.e. Why not define the probability function as p(a<x<b)=integral from a to b of P(x), such that a≠b, and p(a<x<b) is undefined such that a=b, where P(x) is the probability density function?
I suppose, to an extent, this is just a matter of semantics: We can certainly redefine the word “probability” so that the sum of the individual probabilies doesn’t have to equal 1, but instead the integral of the probability density function must be equal to 1, but what’s the point? The way I see it, it doesn’t really make sense to ask the probability of a specific event when we’re dealing with a continuous probability distribution and I don’t see how it would be useful. So why not just leave those probabilities undefined?
Why do we say that probability of an individual event in a continuous distribution is 0
philosophyprobabilityprobability distributions
Best Answer
I'll rephrase your concerns for a slightly different, but very related concept: volume.
The volume of a set of points is the volume integral of 1 over the given set. Using this definition, the volume of a single point in space is 0. Yet it doesn't make sense to say that the volume of a set is 0 if there's still something in it. Furthermore, it introduces the paradox that adding up a bunch of sets with volume 0 gives a set with positive volume, since every set is just a union of sets with a single point in it. So wouldn't it be better to just leave the volume of a point undefined?
Here, the answer should be no, a volume of 0 for point-like sets is perfectly fine. There's also no paradox here, since adding only a countable number of zeroes actually produces zero again: any countable set has volume 0. And adding an uncountable number of numbers is undefined, so the volume of uncountable sets need not be 0, since it's an uncountable union of sets with volume zero, so we can't just add their volumes. The closest thing to an uncountable sum we have is the integral, which actually does give us the desired non-zero volume.
There is no difference between the volume case and the probability case. And I want to stress this: the underlying concept behind volume and probability is exactly the same: both probability and volume are defined via so-called measures, which were specifically defined to model the behavior of volumes. Any intuition you have about volume should be carried over to probability as is, with the single exception that probability must be no larger than 1. But that has no bearing on the question wether single points should have a defined volume/probability or not.