Shuffling Extra Footage (1/3) - Persi Diaconis

let me put the King of Hearts there on the bottom there it is when we shuffle when we're dropping cards well you know half the time after a shuffle the King is going to be on the bottom and half the time it's going to be up from the bottom you know what I mean that this time it's up from the bottom but I might have dropped first from this hand and then it would have been on the bottom that's one sort of step after two shuffles of course it might still be on the bottom but what's the chance that it's on the bottom well half the time if it had two if the first time it had to have stayed on the bottom and the second time it had two so that's one in four that it's on the bottom it could be a various other places but let's just keep track of what's the chance that's on the bottom so after you know three shuffles it's one in eight that it's still on the bottom you'd like that number to be close to one in 52 because when the cards are really mixed up what's the chance the King of Hearts is on the bottom it's one in 52 well let's do the powers of two you know after one shuffle it's a half or two you know on you know it's four six and so two to the two to the sixth is 64 and two to the seventh is 128 so it's sort of in the right ballpark that just to follow one card it's going to take you know six or seven shuffles that makes you can start to see that you know you know because each time some now but of course I'm saying our theorem shows it not only is the King of Hearts right any question about the deck is right you can ask and what's the chance that the ACE is above the deuce that the three is above the seven that there are ten hearts any question is right well that's a lot more questions and it's quite surprising that for any task after seven shuffles seven or eight err no but okay for any task the deck is is close to random I like to say that if you know you were playing with one of your kids maybe you know maybe four or five shuffles is fine if your kid isn't clocking the deck but if the national security depends on it okay shuffle eleven times but but there's an interesting thing that came out of that there's a sharp answer to the question there's a threshold and there's an image that I can try to communicate which is like this suppose I had a big bowl of glass a big glass bowl and it had black balls on the bottom and white balls on top and I had a canoe paddle it's a big bowl and I'm stirring it with my canoe paddle okay at first you know most of the black balls are on the bottom and then this big swaths of black and big swaths of white nothing random about it at all and then all of a sudden it turns gray and then it stays gray you know it stays be mixed well that's what happens with most any mixing process that's again a theorem that is there's a sharp threshold that before that threshold it's this you can see the structure it's not random and after that threshold it's about as random as it can be in it well of course it's never perfectly random it goes to random exponentially fast but it it stays flat and then it dips down and and the it that I'm talking about is the answer to almost any question how many cards can I guess right or if you take some tests for randomness that you like before the seven shuffles or three-halves log to the base 2 of n the the cards that you can you'll fail the test and after no pass it and for any test so so that's sort of it's nice that there's a satisfactory answer you know it might have been that it just sort of dropped smoothly in which case there's no right answer you know this is sort of gets more and more random as you shuffle it does get more and more random but this phase transition which we call the cutoff phenomena happens in most every mixing scheme and so that's made a new area of mathematics people trying to understand you know why is the cutoff phenomena happen and is it can you show that for most mixing schemes that's true when doesn't it happen it doesn't always happen we can find randomization schemes where there is no cutoff but you have to work at it did this change casino policies around the world was this a was this a an earthquake for them well it's it's it'll probably be on my tombstone seven shuffles you know suffice they did change the laws in Nevada so that's that's one man that's one one point and what's to say yeah that's had to be shuffled seven times right and it's seven shuffles plus two of these that you see these are all the same this this you know in the hands or the way they do it in a casino is this way they don't do that you know so they'll they'll they'll strip cut so the laws are they they do seven shuffles and intersperse it with strip cuts that's from 52 cards I'm working on a funny project now which came both from security business cryptography and also from Dungeons & Dragons and crazy things like that I had two colleagues come to me and say how do I shuffle large decks of cards one of my colleagues had a son who plays Dungeons & Dragons type game they have special cards with goodness knows what on them and there are 250 of them and you can't riffle shuffle 250 cards in is there a reasonable scheme for physically mixing up large decks of cards of course in casinos they also with eight decks you can't really shuffle eight decks and and I am I am busy analyzing that with some of my graduate students it's fighting back but we're we're making progress what they do in casinos is they they cut the cards into into groups like this and then they you know shuffle E's pretty well and then shuffle leaves pretty well shuffle leaves pretty well shuffle ease pretty well maybe seven shuffles for each packet and then they will you know take some of these some of these you know and transfer in that way and then shuffle shuffle shuffle leaves you can ask you know how many shuffles how many piles how does it how does it go it's a math problem some of us are working pretty hard on it it's open as of this writing so next next time I hope I answer so when casinos you often see slot machines you know they come no slot machine that is in casino nowadays works with physical randomness gears and timing and stuff like that all of them use computer chips that have pseudo-random number generators in them

Channel: Numberphile2

Views: 105,383

Rating: 4.9638476 out of 5

Keywords: Persi Diaconis, Shuffling, Mathematics (Field Of Study), Casino (Industry), gambling

Id: 1jfm42Qd7Qw

Length: 7min 8sec (428 seconds)

Published: Mon Mar 23 2015

Reddit Comments

[u/DFGdanger]

#PTSOI will be Standard...Battle of Wits reprint confirmed!!!

Too bad in the video he doesn't say a "good" solution, just what casinos do now. If I use casino rules will I be kicked out for gambling?

[u/FoilSiegeRhino]

dungeons and dragons type games that have who knows what on them

hue