Monday, April 20, 2015

Escaping infinitely many arrows

Suppose infinitely many thin arrows are independently shot at a continuous target, with hitting points uniformly distributed over the target. How many arrows would we need to shoot to make it likely that the center of the target has been hit?

Given finitely or countably infinitely many arrows, the probability that the center will be hit is zero. But what if there are as many arrows as points in the continuum? And what if there are more?

I don't know of a good mathematical model for these questions. Standard mathematical probability is defined up to sets of measure zero, and this makes it not useful for answering questions like this. Questions like this seem to make sense, nonetheless, thereby indicating a limitation of our mathematical models. But perhaps that is a mere seeming.


Dagmara Lizlovs said...


You should take up bow hunting. The deer will love you. :-)

Mark Rogers said...

I think this justifies my letting the computer select my lotto numbers as opposed to selecting my own numbers.

Michael Gonzalez said...

Are questions like this affected at all by the Aristotelian sort of notion that continuums are only potentially infinite, and not actually? I mean what if we take the continuous board as a basic element, and consider any potential "points" on it as secondary or derivative notions which do not actually exist?

I may not be asking that correctly...