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Clock Elite user Los Angeles, CA 460 Posts |
If I took a deck of cards (in no particular order) and shuffled it 10 billion times, what are the chances one of those times the deck would be back in perfect new deck order?
Not planning an experiment, just curious what the equation and answer would be... |
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Josh the Superfluous Inner circle The man of 1881 Posts |
Ask this in "Puzzle me this". Some of those guys are scary smart with stuff like that (Nir).
What do you want in a site? "Honesty, integrity and decency." -Mike Doogan
"I hate it, I hate my ironic lovechild. I didn't even have anything to do with it" Josh #2 |
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balducci Loyal user Canada 227 Posts |
Quote:
On 2008-03-01 22:09, Clock wrote: It depends what you mean by 'shuffle'. I think there are 52! = 52 * 51 * 50 * ... * 3 * 2 * 1 ways to arrange a deck. Note very well, the ! denotes 'factorial' not excitement. One of those 52! or 52 factorial ways is in perfect new deck order. So if by 'shuffle' you mean that any of the 52! or 52 factorial ways are equally likely to be obtained after any shuffle, then the probability of the deck being in NDO would be 1 in 52! or 1 in 52 factorial after any such shuffle. I'll continue this discussion with another post in a minute ...
Make America Great Again! - Trump in 2020 ... "We're a capitalistic society. I go into business, I don't make it, I go bankrupt. They're not going to bail me out. I've been on welfare and food stamps. Did anyone help me? No." - Craig T. Nelson, actor.
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balducci Loyal user Canada 227 Posts |
52 factorial is a huge number, equal to approximately 8.065818e+67 (the e+67 means you multiply the number by 10 raised to the power 67).
So 1 in 52 factorial is about as close to 0 probability as you can get. Now, remember I said the answer depends on what you mean by shuffle? If you start with a deck in NDO and perform Faro out shuffles, for example, you know full well that you will get NDO back every 8 Faros. So in that case, the probability would be 1 in 8. So you can see that the answer varies dramatically, depending on what you mean by 'shuffle'.
Make America Great Again! - Trump in 2020 ... "We're a capitalistic society. I go into business, I don't make it, I go bankrupt. They're not going to bail me out. I've been on welfare and food stamps. Did anyone help me? No." - Craig T. Nelson, actor.
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MagicSanta Inner circle Northern Nevada 5841 Posts |
If you do a perfect faro it will be perfect every eight or every 52 shuffles depending on in our out. I just say Balducci mentioned the faro, I was confused by the lack of a link.
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Clock Elite user Los Angeles, CA 460 Posts |
You're right, "shuffle" is hard to justify. I don't mean doing faro's....
Probably more along the line of a random number generator deciding the order of a deck 10 bilion times. New deck order is just an example. Any order that is pre-determined would be the same idea. |
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Clock Elite user Los Angeles, CA 460 Posts |
Thanks balducci...
I think that's it. |
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balducci Loyal user Canada 227 Posts |
Quote:
On 2008-03-01 22:33, balducci wrote: Magicsanta's post prevented me from editing my last post, so let me update it here. 1 in 52 factorial is about as close to 0 probability as you can get. That is the probability of getting NDO in a single shuffle, if you are using one of the totally random shuffles I was talking about. If you do do this 10 billion times, the number of times you would expect to see NDO would be 10 billion multiplied by the probability 1 over 52 factorial. That is still essentially zero. Basically, if you saw even one NDO show up, you would be pretty lucky if you are doing truly random shuffles. As noted before, it is a completely different story if you start with the deck in NDO and perform only Faros.
Make America Great Again! - Trump in 2020 ... "We're a capitalistic society. I go into business, I don't make it, I go bankrupt. They're not going to bail me out. I've been on welfare and food stamps. Did anyone help me? No." - Craig T. Nelson, actor.
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Clock Elite user Los Angeles, CA 460 Posts |
Definitely.
This is kind of cool to think about... A deck of cards that you have around the house that has been well used and shuffled is most likely to be in an order that has NEVER been seen before in the history of playing cards. |
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Clock Elite user Los Angeles, CA 460 Posts |
The reason is this...
NDO is just like any other well mixed deck of cards... just another 1 in something trillion of combinations. |
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Clock Elite user Los Angeles, CA 460 Posts |
You have a 1 in 80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 chance of shuffling a deck of cards in ANY particular order.
Lottery odds are about 1 in 120,000,000. This makes the lottery sound like childs play!! So any well mixed deck is very easily a "first" time order in the history of cards. |
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Doug Higley 1942 - 2022 7152 Posts |
BUT and this is where the ALL the odds go cablooey...if it CAN be done in
80,658,175,170,943,878,571,660,636,856,403,766,975,289,505,440,883,277,824,000,000,000,000 times then it can be done the FIRST time or the 10th. It is why a single entry in a contest of 300,000,001 entries with all but the one put in by one other person can win over that person. It falls to luck and 100% chance without any odds.
Higley's Giant Flea Pocket Zibit
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MagicSanta Inner circle Northern Nevada 5841 Posts |
There are only x amount of ways a deck can be stacked, that isn't your question. Let us look at the odds of five specific cards being in specific spots in a deck during a poker game, the royal flush. This will occur 4 times out of the possible 2,598,960 possible hands that can be dealt. That being said I think the answer is officially "who cares?". The reason I say this is there is always a chance that you can shuffle a deck and eventually have it match a shuffle that existed before, the 'odds of high numbers' means you may stumble into a stack you wanted to hit. The problem is when using imperfect shuffles which every shuffle is different Dr. Diaconis, a man smarter than all of us, and this was verified on a computer experiement and challenged by academics, is that it takes seven shuffles of a deck to have a random and complete shuffle. Anything less than seven isn't enough and anything more, to quote the report by Perci:
"IT takes just seven ordinary, imperfect shuffles to mix a deck of cards thoroughly, researchers have found. Fewer are not enough and more do not significantly improve the mixing. " Since you don't have a specific interaction of the cards you cannot measure to see at what point you hit the specific stack. Each imperfect shuffle changes the numbers. The calculation as to getting one card blindly to a specific spot taking in all imperfect shuffles is 0.8 X 10 to the 68th power. The statement that you can shuffle a deck of cards and it will be in a stack that has never been seen before in history is, due to the huge number of cards and the number of times they are shuffled worldwide means that pretty much every possible combination has been hit. Heck I've been in poker tourniments where two royal flushes where hit on separate tables in half an hour and I've not seen one since, but I would think someone gets one someplace every day. |
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MagicSanta Inner circle Northern Nevada 5841 Posts |
Doug is absolutely right by the way. I use to see something unusual and someone would say "man! What were the odds of that happening?" and I'd reply "seemingly based on what we just saw 100%".
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James F Inner circle Atlanta 1096 Posts |
From: http://www.matthewweathers.com/year2006/shuffling_cards.htm
"When you multiply all those numbers together, you get 80658175170943878571660636856403766975289505440883277824000000000000. That number is 68 digits long. We can round off and write it like this: 8.0658X1067. How many times have cards been shuffled in human history? That's an impossible number to know. So let's overestimate. Currently, there are between 6 and 7 billion people in the world. Also, the modern deck of 52 playing cards has been around since 1300 A.D. probably. If we assume that 7 billion people have been shuffling cards once a second for the past 700 years, that will be way more than the actual number of times cards have been shuffled. 700 years is 255675 days (plus or minus a couple for leap year centuries), which is 22090320000 seconds. Now, if 7000000000 people had been shuffling cards once a second for 22090320000 seconds, they would have come up with 7000000000 • 22090320000 different combinations, or orderings of cards. When you multiply those numbers together you get 154632240000000000000, or rounding off, 1.546X1023. So, it's safe to say that in human history, playing cards have been shuffled in less than 1.546X1023 different orders. Is this order unique in human history? Probably so. When I shuffled the cards this afternoon, and came up with the order you see in the picture, that is one of 8.0658X1067 different possible orders that cards can be in. However, in the past 700 years since playing cards were invented, cards have been shuffled less than 1.546X1023 times. So the chances that one of those times they got shuffled into the same exact order you see here are less than 1 in 100000000000000000000000000000000000000000000 (1 in 1044)." ...Absolutely insane. James |
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balducci Loyal user Canada 227 Posts |
Quote:
On 2008-03-03 11:02, James F wrote: Obviously just a typo, but that last line should say 1 in "10 to the 44th power", not 1 in 1044.
Make America Great Again! - Trump in 2020 ... "We're a capitalistic society. I go into business, I don't make it, I go bankrupt. They're not going to bail me out. I've been on welfare and food stamps. Did anyone help me? No." - Craig T. Nelson, actor.
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MagicSanta Inner circle Northern Nevada 5841 Posts |
He may not have ben able to put the tiny 44. Look up the study done by Perci Diaconis (anyone know if he is related to Jim Diaconis?) on the mathmatics of shuffling cards.
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James F Inner circle Atlanta 1096 Posts |
Quote:
On 2008-03-03 13:14, MagicSanta wrote: Yeah, if you click on the link (where I got all that from), you can see it actually shows 10^44 but I guess it messed up with the copy and paste. But it's good to point that out so no one misunderstands it as 1044. James |
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ClintonMagus Inner circle Southwestern Southeast 3997 Posts |
I KNOW how many times cards have been shuffled in human history, but I'm not telling anyone.
Oops... there's another one!
Things are more like they are today than they've ever been before...
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MagicSanta Inner circle Northern Nevada 5841 Posts |
I was at a poker tourniment (you wouldn't believe the bad players that join up) and there was lots of shufflin' just there.
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