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Subject:  Re: 5 Statistics Problems To Change Your View Date:  11/27/2012  11:11 AM
Author:  LorenCobb Number:  409725 of 578555

A couple of days ago PolymerMom posted an excellent set of five classical problems "that will change the way you see the world". Pondering these kinds of paradoxes and counter-intuitive situations is great for the mind -- one can almost always learn something. In reply, I posted another set of three paradoxes, also from probability and statistics, which have in recent years achieved a considerable degree of notoriety in mathematics. From the meagre responses it is clear that I employed language and concepts that are foreign to most people, and so I owe an apology to the board. It was not my intention to mystify, only to show how easily paradoxes arise in the realm of the infinite as well as finite probabilities. These too can "change the way you see the world", but only if you have some experience with the Axiom of Choice.

FWIW, here are some answers, with links for the curious.

Q1. (Lebesgue measure) In measure theory, and hence in probability and statistics, the "Lebesgue measure" of the interval from zero to one is just its length, i.e. 1. Similarly, the Lebesgue measure of a unit square is just its area, also 1. The Lebesgue measure of a unit cube is just its volume, again 1. The Lebesgue measure of a unit hypercube of, say, N dimensions is also 1, for any finite N. What is the Lebesgue measure of a hypercube of infinite dimensions?


Why zero? Here is a cute but informal argument: Imagine for a moment that the Lebesgue measure of a hypercube of (countably) infinite dimensions has a positive measure, say one. Since the hypercube is infinite-dimensional, it can be carved a (countably) infinite set of pairwise disjoint identical hypercubes, each one with side length one-half that of the original. Since the sum of the volumes of this infinite set of smaller hypercubes is finite, each of the smaller hypercubes must have measure exactly zero. But the volume of the original hypercube is the sum of its parts, so it too must have volume zero. This is a contradition, therefore the original hypercube cannot have a positive measure.

Wikipedia article:

Q2. (Vitali set) We can all agree, I think, that if we throw a dart at a dart board in a completely fair way -- so that all zones of the same size are equally likely to be hit -- then the probability of hitting any particular zone is proportional to the size of the zone. Okay, let's construct a paradoxical dart board:

Start with a dart board that is just a straight line of length one. Paint it red. Now divide the dart board into zones so that two points are in the same zone if they differ by a rational amount (an amount expressible as a ratio of two integers). From each zone, pick a single point and paint it black.

You grab a dart whose point is sharpened so well that it is truly just a point, not a surface. You throw this dart at the board, in a completely fair way, and it hits the board. What is the probability that the point of your dart has hit a black point?

A2: The probability cannot be determined. It does not exist.

Why does this probability not exist? Because the set of black points does not have a measure. The "Vitali Set", which this puzzle describes, is the first-discovered (1905) and best-known example of a non-measurable set. The dart board is also non-constructive -- meaning you cannot go out and make one -- because it relies on the infamous Axiom of Choice (picking one each from an uncountable set of sets [cue evil music here]). We can conclude from this exercise that probabilities, even simple ones based on throwing darts, are not as straightforward as they appear.

Wikipedia article:

Q3. (The Banach-Tarski paradox) Consider a solid sphere in three dimensions. For motivation, it may help to imagine that it is solid gold. Is there a way to dissect the sphere into a finite number of non-overlapping pieces, and then to reassemble them into TWO spheres of exactly the same size and volume as the first? The reassembly process must use only moving the pieces around and rotating them, without changing their shape.

A3: YES, there is a way to do it.

Alas NO, you cannot write an algorithm to achieve it, or design a machine that will do it, because the manufacture relies on the Axiom of Choice [more evil music], which is non-constructive. The Banach-Tarski paradox remains an intriguing and active research area in mathematics.

Layman's guide to Banach-Tarski:
Wikipedia article:

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