Suppose you throw a dart at a circular target of some radius r in such a way that the possible impact points are uniformly distributed over the target. Classically, the probability that you hit the center of the target is zero. But suppose that you believe in infinitesimal probabilities, and hence assign an infinitesimal probability α(r)>0 to hitting the center of the target.
Now, α(r) intuitively should vary with r. If you double the radius, you quadruple the area of the target, and so you should be only one quarter as likely to hit the center. If that’s right, then α(r)=β/r2 for some infinitesimal constant β.
This means that in addition to the usual constants of physics, there is a special infinitesimal constant measuring the probability of hitting the center of a target. Now, there is nothing surprising about general probabilistic stuff involving constants like π and e. But these are dimensionless constants. However, β is not dimensionless: in SI units, it is expressed in square meters. And this seems incredible to me, namely that there should be a non-dimensionless constant calibrating the probabilities of a uniformly distributed dart throw. Any non-dimensionless constant should vary between worlds with different laws of nature—after all, there will be worlds where meters make no sense at all (a meter is 1/299792458 of the distance light travels in a second; but you can have a world where there is no such thing as light). So, it seems, the laws of nature tell us something about the probabilities of uniform throws. That seems incredible.
It is so much better to just say the probability is zero. :-)
4 comments:
The intuition that alpha(r) should change with r can be undercut. Suppose you increase the target size but correspondingly increase the distance from the target, and suppose there is no gravity or wind. Then the same angle hits the center in both cases, so alpha(r) doesn't change.
Apart from ensuring regularity, what would infinitesimal probabilities be good for? Maybe to answer questions like this: given that the dart hit either point A or point B, what is the probability that it hit A? For such purposes, only ratios matter. Also, infinitesimals are by definition incommensurable with reals (in an Archimedean sense), so surely only ratios could matter.
Maybe also gambling with infinite utilities? "Do you take this infinitesimal chance at this eternal bliss, or a certainty of a cupcake?"
I doubt that there could be a definite answer in that sort of case. But how about this? There is a fair infinite lottery. If the outcome is N, you have an F(N) chance of winning $G(N). For some F and G (e.g. both constant), the setup has a well-defined value. But what if, for example, F(N) = 1/N, G(N) = N? Or F(N) = N mod 2, G(N) = constant?
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