P(X = exact value) = 0: Is it really counterintuitive?

by lucidfox 1 min read29th Jul 201149 comments

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I'm probably not going to say anything new here. Someone must have pondered over this already. However, hopefully it will invite discussion and clear things up.

Let X be a random variable with a continuous distribution over the interval [0, 10]. Then, by the definition of probability over continuous domains, P(X = 1) = 0. The same is true for P(X = 10), P(X = sqrt(2)), P(X = π), and in general, the probability that X is equal to any exact number is always zero, as an integral over a single point.

This is sometimes described as counterintuitive: surely, at any measurement, X must be equal to something, and thus its probability cannot be zero since its clearly happened. It can be, of course, argued that mathematical probability is abstract function that does not exactly map to our intuitive understanding of probability, but in this case, I would argue that it does.

What if X is the x-coordinate of a physical object? If classical physics are in question - for example, we pointed a needle at a random point on a 10 cm ruler - then it cannot be a point object, and must have a nonzero size. Thus, we can measure the probability of the 1 cm point lying within the space the end of the needle occupies, a probability that is clearly defined and nonzero.

But even if we're talking about a point object, while it may well occupy a definite and exact coordinate in classical physics, we'll never know what exactly it is. For one, our measuring tools are not that precise. But even if they had infinite precision, statements like "X equals exactly 2.(0)" or "X equals exactly π" contain infinite information, since they specify all the decimal digits of the coordinate into infinity. We would have an infinite number of measurements to confirm it. So while X may objectively equal exactly 2 or π - again, under classical physics - measurers would never know it. At any given point, to measurers, X would lie in an interval.

Then of course there is quantum physics, where it is literally impossible for any physical object, including point objects, to have a definite coordinate with arbitrary precision. In this case, the purely mathematical notion that any exact value is an impossible event turns out (by coincidence?) to match how the universe actually works.

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