現代数学の系譜工学物理雑談古典ガロア理論も読む67

[過去ﾛｸﾞ] 現代数学の系譜工学物理雑談古典ガロア理論も読む67 (1002ﾚｽ)
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298: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/06/08(土) 18:35:14.65 ID:e2T0R87W

>>294 参考
＞P75に、「キューブ工場」という確率のパラドックスがある

（参考）
http://www.untrammeledmind.com/2018/01/three-strange-results-in-probability-cognitive-states-and-the-principle-of-indifference-monty-hall-flipping-coins-and-factory-boxes/
UNTRAMMELED MIND
IDEAS IN PROGRESS

JANUARY 1, 2018 BY DAN JACOB WALLACE
Three Strange Results in Probability: Cognitive States and the Principle of Indifference (Monty Hall, Flipping Coins, and Factory Boxes)
(抜粋)
Probability is known for its power to embarrass our intuitions. In most cases, math and careful observation bear out counterintuitive results.
After many such experiences, one’s intuition improves (sometimes perhaps crossing into a kind of overcorrection?see the Optional Endnote for some inchoate thoughts on that).
But some results stay strange, and it’s not always clear whether our rebelling intuitions signal a problem with formal probability, or simply confirm that human cognition has evolved to concoct tidy stories amounting to illusory?if sophisticated?representations of the world rather than to deal head on with complexity, chance, and uncertainty.

(3) Factory Boxes and the Principle of Indifference: The principle of indifference, also known as the principle of insufficient reason, says that when you see no reason to weight competing outcomes differently, you should weight each of them as equally probable. I gave examples in (2) above.
The most common application might be when we assume a given coin is fair.

つづく

http://rio2016.5ch.net/test/read.cgi/math/1559830271/298

300: 現代数学の系譜 雑談 古典ガロア理論も読む ◆e.a0E5TtKE  [] 2019/06/08(土) 18:37:36.38 ID:e2T0R87W

>>298

つづき

Bas van Fraassen has produced a compelling paradox arising from this principle.2 Here I’ll quote Aidan Lyon’s discussion in his 2010 paper “Philosophy of Probability” (published as a chapter in Philosophies of the Sciences: A Guide):

Consider a factory that produces cubic boxes with edge lengths anywhere between (but not including) 0 and 1 meter,
and consider two possible events:
(a) the next box has an edge length between 0 and 1/2 meters or
(b) it has an edge length between 1/2 and 1 meters.
Given these considerations, there is no reason to think either (a) or (b) is more likely than the other,
so by the Principle of Indifference we ought to assign them equal probability: 1/2 each.
Now consider the following four events:
(i) the next box has a face area between 0 and 1/4 square meters;
(ii) it has a face area between 1/4 and 1/2 square meters;
(iii) it has a face area between 1/2 and 3/4 square meters;
or
(iv) it has a face area between 3/4 and 1 square meters.
It seems we have no reason to suppose any of these four events to be more probable than any other, so by the Principle of Indifference we ought to assign them all equal probability: 1/4 each.
But this is in conflict with our earlier assignment, for (a) and (i) are different descriptions of the same event (a length of 1/2 meters corresponds to an area of 1/4 square meters).
So the probability assignment that the Principle of Indifference tells us to assign depends on how we describe the box factory: we get one assignment for the “side length” description, and another for the “face area” description.
(引用終り)

http://rio2016.5ch.net/test/read.cgi/math/1559830271/300

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