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

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.