Suppose that a countably infinite number of infinitely thin or perfectly symmetrical arrows is independently shot at a continuous target, with the distribution of impact points uniform over the target. (The independence requires that the arrows can go through each other--say, because they are made out of laser beams--or are removed between shots.) How probable is it that the exact center of the target will be hit by at least one arrow? In classical probability, the answer is zero. Intuitively, this is because the number of points on the target is a bigger infinity than the countably infinite number of arrows.
What if, instead, the number of arrows is greater than or equal to the number of points on the target? Unfortunately, the standard probabilistic model (a product space with the number of factors equal to the number of arrows) for the situation cannot answer that question: the probability of a point being hit by at least one of an uncountable infinity of arrows will be undefined. It would be interesting to see if there is any way of getting a non-arbitrary answer to the question outside of the standard model, say by putting some natural restrictions on which extensions of the model one allows.
7 comments:
How about this? A density D of arrows means by definition Poisson(D) hits at each point, independently for each point. But this may not be what you have in mind.
No, that's not what I had in mind. What I had in mind is basically this: Take an uncountable collection of independent random variables uniformly distributed over [0,1]. Is there a canonical answer to the question: What is the probability that NONE of them takes the value 0?
This is not meant to be a mathematically precise question. Basically, the question is: How to make this a mathematically precise question?
There are at least two difficulties with the question.
1. The event "None of the variables takes the value 0" is not measurable in the product measure when the number of variables is uncountable.
2. The mathematical understanding of a "uniform" distribution is not sensitive to sets of measure zero, like { 0 }. Thus, you can have a variable that counts as having a uniform distribution on [0,1] but that is guaranteed not to take the value 0 (or not to take a rational number value, etc.). And no matter how many independent variables guaranteed not to take the value 0 you have, it's still guaranteed that none of them take the value 0.
That many arrows grouped so tightly is awfully hard on the fletching. :-)
And it's guaranteed that there will be some Robin Hoods.
It's even more of a problem with my Galeforce crossbow and the bolts I use. I have to have use several different points on the target when shooting because they group so tightly. This is with field points. Hunting. broadheads might yield a different result.
How many arrows would it take to demonstrate this to a confidence level of 90%?
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