Wednesday, November 7, 2012

A conditional probability argument for zero probability events

Suppose a dart is tossed at a circular target centered on the point 0, in such a way that it has uniform probability of landing anywhere within a meter of 0, and cannot land anywhere else. Let Z be the event that the dart lands at 0. Let E be the event that the dart lands within half a meter of 0.

Suppose, for a reductio, that P(Z)>0. Presumably, P(Z) is going to be some infinitesimal. Now consider P(Z|E). Intuitively this should be the same as the probability P2(Z) that a second dart that has uniform probability of landing within half a meter of 0, and unable to land anywhere else, lands at 0. But I claim that P2(Z)=P(Z). For surely when you have a uniform distribution over a target, and you scale the target up or down, the probability of landing at the center should remain the same. But now P(Z|E)=P2(Z)=P(Z). But if P(Z)>0 then P(Z|E)=P(ZE)/P(E)=P(Z)/P(E). Moreover, P(E)=1/4 (since E has a quarter of the area of the radius 1 circle). Thus P(Z|E)=4P(Z). Thus 4P(Z)=P(Z). And this can only be true if P(Z)=0.

Suppose you challenge my scaling invariance claim. Then you will have to think that there is a non-constant mathematical function f from lengths to probabilities such that f(r) is the probability of hitting the exact center of a target of radius r. The above argument then shows f(1/2)=4f(1). Generalizing the above argument, we can see that f will have to satisfy the formula f(ar)=f(r)/a2 for all a>0.

But what determines this function? The distances that are plugged into f are genuine-distances-with-units. Units for distances depend conceptually on laws of nature: think of the Planck Length or of the meter, which is 1/299792458 of the distance light travels in a vacuum in a second. Does f then depend on the laws of nature in some empirically nonverifiable way? And isn't it amazing that there be this function from distances to infinitesimals that tells us what exactly the probability of hitting the center of a target of one meter is?

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