Post #401,952
5/21/15 6:44:37 PM
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Fun with statistics
Let's say you're the dealer at a casual Friday night poker game. Let's also say, for the sake of argument, that you're an expert shuffler, and not one of those people who just clumsily swirls cards around like an infant. You expertly riffle the cards, toss them hand to hand, juggle them, throw them into a hat, etc., until eventually you're confident that you've fully randomized the cards.
What are the chances that the configuration of the deck you now hold is the same as one that you've shuffled before on a previous poker night? One chance in 1,000? One in 10,000? We mean, there's only 52 cards, so how many can it really be?
You should feel special, because it's almost certain that the configuration of the deck you hold in your hand has never been held by any human being in the history of mankind, on this Earth, or on any one of its many parallel universes. You currently hold in your hand something that will never again be seen, from now until the end of time itself.
It's true that 52 cards doesn't seem like a lot. But if you try to count the number of possible combinations of those cards, you better have a few evenings free. The total number of statistical combinations of a 52-card deck is what's known as "52 factorial," sometimes referred to as "52!" or "52 shriek." Written out in full, that number is:
80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000
That's a giant-ass number. To put into into perspective, it's been calculated that "if every star in our galaxy had a trillion planets, each with a trillion people living on them, and each of these people has a trillion packs of cards and somehow they manage to make unique shuffles 1,000 times per second, and they'd been doing that since the Big Bang, they'd only just now be starting to repeat shuffles."
http://www.cracked.com/article_22432_6-bizarre-statistics-that-prove-math-black-magic.html
Satan (impatiently) to Newcomer: The trouble with you Chicago people is, that you think you are the best people down here; whereas you are merely the most numerous.
- - - Mark Twain, "Pudd'nhead Wilson's New Calendar" 1897
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Post #401,953
5/21/15 7:09:51 PM
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[wibble]
To put it in perspective, if this impossible thing times this impossible thing times this utterly inconceivable thing ...
His sense of perspective sucks.
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Post #401,956
5/21/15 8:42:06 PM
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as long as the cards end up where I want them, shrug
Any opinions expressed by me are mine alone, posted from my home computer, on my own time as a free American and do not reflect the opinions of any person or company that I have had professional relations with in the past 59 years. meep
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Post #401,959
5/21/15 11:09:20 PM
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Statistics and probability can be fun.
Also counter-intuitive and mind-bending in ways that can be entertaining for onlookers.
I recently did some analysis of this sort of problem: Roll a dice. Keep rolling until you get a 1. How long are you likely to be doing that? Now make that 1 in 100. Or 1 in 1000.
Not surprisingly, the probability curve is the same shape, but it drops slower for the larger range. For 1 in 10, you have a 50% chance of rolling a 1 by the fifth roll. For 1 in 100, it's the 68th roll. But in both cases, the probability of it going ten times-longer is about 0.01%. :-)
Wade.
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Post #401,968
5/22/15 1:30:14 AM
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Feh, such a tiny inconsequential number
You want a bit of Graham's Number, you do. the observable universe is far too small to contain an ordinary digital representation of Graham's number, assuming that each digit occupies one Planck volume. Although I must confess that I would have never happened across Graham's Number, myself, as the following question has never raised itself in my mind: Connect each pair of geometric vertices of an n-dimensional hypercube to obtain a complete graph on 2n vertices. Colour each of the edges of this graph either red or blue. What is the smallest value of n for which every such colouring contains at least one single-coloured complete subgraph on four coplanar vertices?
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Post #401,969
5/22/15 5:33:35 AM
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I was thinking about Graham's Number, too.
Though there does seem to be a kind of insanity to how you have to write it.
Wade.
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Post #401,971
5/22/15 8:36:58 AM
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Who needs to be constrained by the Universe?
Kruskal's Tree Theorem: The latter theorem ensures the existence of a rapidly growing function that Friedman called TREE, such that TREE(n) is the length of a longest sequence of n-labelled trees T1,...,Tm in which each Ti has at most i vertices, and no tree is embeddable into a later tree.
The TREE sequence begins TREE(1) = 1, TREE(2) = 3, then suddenly TREE(3) explodes to a value so enormously large that many other "large" combinatorial constants, such as Friedman's n(4),[*] are extremely small by comparison.[1] A lower bound for n(4), and hence an extremely weak lower bound for TREE(3), is A(A(...A(1)...)), where the number of As is A(187196),[2] and A() is a version of Ackermann's function: A(x) = 2 [x + 1] x in hyperoperation. Graham's number, for example, is approximately A^64(4) which is much smaller than the lower bound A^A(187196)(1). It can be shown that the growth-rate of the function TREE exceeds that of the function fΓ0 in the fast-growing hierarchy, where Γ0 is the Feferman–Schütte ordinal. Neat stuff. Cheers, Scott.
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Post #401,991
5/23/15 1:44:56 AM
5/23/15 2:15:32 AM
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I suspect that Monty Python's send-up of these utterly surreal imaginations
was the skit with the aged author/artist? sitting outdoors in a chair, as the reporter repeatedly refers to him by his full-name: {some thousands of mouth-noises long} ..as the subject finally expired amidst the final recitation: a tribute turned into a threnody, much like MAHLER: "Das irdische Leben" ("Earthly Life")
[German text from the anthology Des Knaben Wunderhorn (The Youth's Magic Horn)] where the child dies as his mutter repeatedly tells him to wait-a-while ... as the bread bakes. I suppose that such ruminations may so occupy the grey cells in some homo saps as to preclude their otherwise emulating Dr. Moriarty or other fictional Mr. Hyde-grade outliers, becoming just another nasty menace as they age. (So this may be a good thing, in such cases.) But I fail to see how such MIne's Bigger exercises can take precedent over the efforts needed next by millions of the math-unafflicted: in mere repetitive, pedestrian labors towards planetary rescue from [Twain] The whole damned human race {/Twain] Do. That. ... and there will be time for mental masturbations, again. I wot.
ie We are not amused.
PS: Believe n! has been mentioned; it is al ye Need to know in imagineering such tempests in academic teapots; Teller was right: most people Fail in perceiving emotionally the consequences of exponentials (and n!) ... Now There is.. applied math for a species with short attention-span.
;^>
Edited by Ashton
May 23, 2015, 02:15:32 AM EDT
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