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Subject:  Re: 5 Statistics Problems To Change Your View Date:  11/25/2012  3:16 PM
Author:  LorenCobb Number:  409553 of 567685

PM: This is a thought-provoking article to take to heart...

Yes, indeed. Having been a consulting statistician for, um, the past forty years, I had heard these old chestnuts many times. Except for #3 (Abraham Wald's Memo), that is, which was new to me -- and I loved it.

The bedrock foundation of modern probability and statistics is something called "measure theory", which has its own collection of counter-intuitive puzzlers. Your readers may enjoy testing their mettle against these as well. It helps to know that a probability is just a special case of a measure.

These head-scratchers are arranged in order of ascending difficulty. The answer to the third could make your fortune!

1. (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?

2. (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?

3. (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. [Hint: in this puzzle, size = volume = measure]

Answers will appear here on Monday night.

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