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Server Time: 12/1/2008 9:56:38 PM PACIFIC |
 Really big numbers, stdioh, 8. Aug 2003 16:14 | ||
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| This is a response I posted to another thread, but I am reposting here because I think it is important to understand how really big some quantities are. At issue is the likelyhood that 2 decks were ever configured the same way after a proper shuffle in the history of the world. There are 52! equally likely deck configurations. That number is really really really big. _____________________________ no no no no no. It seems that you're falling victim to a classic piece of innumeracy here. Not understanding how really damn hell big some gigantic numbers are. There are 365 (really 366) birthdays and 30 people. The chance of them sharing a birthday is obviously within the bounds of understanding for most smart people. Now lets look at how many poker hands there have ever been, but I'm going to really stretch things for you. Lets assume that there are an average of 100,000 poker tables in operation on planet Earth, all of them playing around the clock. Lets say that they each deal an increadible 50 hands per hour so you're getting in 5 million hands per hour worldwide. Then you get 120 million hands per day or about 43.8 billion hands per year. Lets say that they play for 1000 years (much longer than we've had cards at all) then you get 43.8 trillion poker hands dealt and one fresh shuffle for each deal. Sounds like a lot, huh? I'll bet that you could definitely have repetition in your 52! with that many, right? 43.8 trillion is much less than 17! ... it is massively dwarfed by a number so giant as 52! What if they played for a million years? They would still play much less than 19! hands. Better yet, lets take 6 billion people (more than there are on earth) and have them shuffling for 1 million years each at 10 shuffles per minute. You get 5,256,000 shuffles per person per year, so you get 31.536 quentillion hands dealt. How much is that really? Very much less than 24! Pretty soon we're going to need to enlist genetically modified super ants to shuffle for us. 24! is so tiny compared to 52! that it is as insignificant as a gnat flying around the arse of a cow standing by the highway that you drive by at 120 km/h ten years from now. For the sake of humour, the universe is about 9 billion years old. If you took your 6 billion continuous shufflers and they had been going since the beginning of time, until last Tuesday, you would have still less than 26! total shuffles. So the long and short of it is that it is very different from Slim's husstle. It is so unlikely that there were ever two decks shuffled into the same position (assuming reasonable and fair shuffling of course) that there is a greater likelyhood that our sun has already exploded and we'll know about it within the next 9 minutes. | ||
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| Re: Really big numbers, pt_Gatsby, 8. Aug 2003 16:29 | ||
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| :) That one is probably for me. You are correct though, I definately didn't appreciate the size of 52! until I pulled out the old "what are the chances of", and ran the sampling with 40 billion shuffles... All I have to say is that it's quite humbling to run through p(n,d) ~ 1-e^(-n(n-1)/2d) That's what it took to convince me, anyway. Never again will I argue that... Its just hard to accept, I guess! | ||
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| Re: Really big numbers, johnph77, 9. Aug 2003 14:00 | ||
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| Agreed. And even if you run down through the mathematical possibilities of duplicating the stack to the point where the possibility of any duplication equals 50% - formula: 1/(52!) +1/(52!-1) +1/(52!-2)... - they're infinitesmal. But it can happen..... | ||
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| Re: Really big numbers, johnph77, 9. Aug 2003 11:44 | ||
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| The old New Jersey Lotzee game - and I don't know when it was changed - gave the bettor a choice of four numbers from 00 to 99 on 21 different levels. On one level, this gave the bettor 3,921,225 different possibilities. Taking this number to the 21st power, there would have been a number approaching 3 followed by 138 zeroes different possible tickets available to the bettor. | ||
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| Re: Really big numbers, johnph77, 13. Aug 2003 13:38 | ||
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| If anyone has played the Minesweeper game provided with Windows, the intermediate game offers: 85,606,851,036,149,690,093,665,078,729,467,284,944,345,648,330,100,234,810,269,826,710,742,855,360,817,240,473,600,000,000,000 different mine placement possibilities..... I'm not even getting into the Advanced game..... | ||
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| Re: Really big numbers, stdioh, 13. Aug 2003 15:35 | ||
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| I think that number would be slightly reduced since there are limitations on where the mines can be placed. Mines are only placed after you have clicked on an initial square so there is a huge division of possibilities there - the first one never has a mine. Also, any board which allows a 3X3 block of mines either with or without a single hole in it is illigal (no "8" squares) ... that divides things again. | ||
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| Re: Really big numbers, johnph77, 13. Aug 2003 16:49 | ||
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| Tis true - that number includes the 216 possibilities where all the 40 mines are placed in a 5x8 block. But the "8" square does exist - I've seen it in custom games. But the number is and was meant to be theoretical and was a result of my having too much time on my hands once.... | ||
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| Re: Really big numbers, stdioh, 13. Aug 2003 17:38 | ||
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| Can it be created in a standard game though? I've never ever ever seen one and minesweeper was practically my religion at one point. My fastest expert time was better than most people can do on medium. My time was 71 seconds. Of course nowadays half my games don't break 100. | ||
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| Re: Really big numbers, johnph77, 13. Aug 2003 18:40 | ||
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| I would suppose that random mine placement would make it possible, but I've never seen it, and rarely see a 7. One of these days I'll work out the odds. My best time in the intermediate game is 72 seconds, and I think the world record NOT using computer solution programs is 61 or 62. | ||
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| Re: Really big numbers, stdioh, 13. Aug 2003 20:53 | ||
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| It is all about having a reliable and familiar mouse that is fine tuned just so. I think that playing minesweeper is very similar to solving a rubics cube. Once you get the theory down to the point that you don't have to think about it anymore it becomes a question of familiarity with the hardware, speed practice, and dexterity. But you can still tweak your firmware to get there just a little faster when you are in practice. Now I'm thinking about taking up cubing again. | ||
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| Re: Really big numbers, johnph77, 14. Aug 2003 01:05 | ||
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| lol - see what I started? gl. | ||
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