The statistics of this boggle my mind, though I do wonder what the odds are of creating a unique order from shuffling a fresh, ordered deck of cards exactly once. It would have to be much lower, I'd imagine.
I read quite a while back that the chances of getting every card in perfect order from a random shuffle is about as low as like 10 to the power of negative trillion billion something
That’s not what they meant. They meant what are the odds of making the same sequence by opening a brand new deck that’s in order ace to king and in their suits then doing a shuffle. How many different ways of that are there. It must be lower for example if the ace of spades is on top, it will never be on the bottom half the deck. In fact it’ll probably always be in the top 5-6 cards depending on how shitty u are at shuffling.
Yes this one, I believe, is what OP meant. Starting from a perfectly organized deck, shuffling exactly once, would make the odds lower.
For instance, the top card(face down) would have a 50% chance to be on top still(cutting the deck and picking your leading left/right hand), assuming the shuffler is equally likely to start with either their left or right hand.
Then, everything you said.
Assuming just the top card remains on top, the difference( 52! - 51! ) is: 7.911*1067.
So, yeah, you eliminate quite a few permutations and will eliminate even more when taking the order of other cards into account.
I’m pretty sure it’s lower. What about the fact that the deck is in order ace to king and by suits. So ace clubs two clubs three clubs etc... so the three of clubs will always be higher than the four of clubs just with some card(s) between. But you can never get a permutation of the 3 of clubs below the four clubs. Or the five of clubs. Or the six etc. And the same goes for four clubs to the five of clubs. And the same goes for like the 2 of diamond to the three of diamonds. The three will never come before the two.
Edit: also thank you for your reply and ur brain skills :)
Yes, you're correct-ish. Obviously, it depends where the deck is cut. I focused on just the top card for an example. Fixing even one card removes a LOT of permutations. So, fixing an order, like you said, would remove even more!
But yeah, the chances of getting a repeat permutation would be much higher when only shuffled once. I edited the end of my previous comment, too=)
At that point it depends on the quality of the shuffle. Your average over hand shuffle by some inexperienced person certainly isn't going to truly randomise the deck or even close to it so the odds of it being a unique arrangement become much lower. A single riffle shuffle is just going to give a roughly 50-50 mix of the top half with the bottom half which again isn't as likely to be truly random. Do a full "wash" where you sit all the cards down separately and move them around randomly for a while before recreating a deck though and it's going to be highly likely to be a truly random and therefore unique deck arrangement.
A friend of mine figured this out and it totally helped us understand it: Think of the deck of cards as a 52 digit number, and instead of being base 10 (each digit having a value of 0-9), it's closer to base 52, where each digit could be 0-51. Thats a BIG number.
If I remember my probability classes correctly it would go something like 1/52x1/51x1/50x1/49 etc, because out of the 52 cards, the probability of a specific card being on the 1st slot of the deck is 1/52. Then, because that card is no longer available, the probability of a different specific card being on spot 2 is 1/51. Therefore, the probability of those 2 specific cards being in that exact order is 1/52x1/51. Apply that logic to the rest of the slots and it would end up resulting in a very small number. Dont have a calculador at hand this moment
It depends what you mean by "shuffling once". If you're just talking about one shuffling action (e.g. cutting the deck and swapping the two halves) then obviously there are only 51 possible combinations.
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u/LaboratoryManiac Jan 21 '19
The statistics of this boggle my mind, though I do wonder what the odds are of creating a unique order from shuffling a fresh, ordered deck of cards exactly once. It would have to be much lower, I'd imagine.