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Diaconis on Randomness
Posted By: Phil Simborg
Date: Wednesday, 22 May 2013, at 8:08 p.m.
In Response To: Diaconis on Randomness (Bob Koca)
I have spent hours (in my previous life) studying the shuffling of the cards, both to see how they are randomized and to see if I could use the information to my advantage in the next hand. I have also studied how to cheat and stack the deck so that I could prevent being cheated, and I must tell you that I have left several games because of the way the cards were being handled by one or more of the players. In some games, I have gone so far as to do what I believe my opponent is doing crooked, but I do it in the most obvious way and call their attention to it to make sure they know I am on to them. If they were innocent, and just happen to be shuffling in the same manner a cheater does, then it goes over their head.
Now, back to "perfect" shuffling. If you NEVER cut the cards and shuffle perfectly every time, putting 26 in each stack and making sure the cards are mixed alternating one from each stack, at the end of 7 shuffles you will see a definite pattern in the cards. They are not randomized. Most of the picture cards and aces will be bunched together, in four regular bunches, equally spaced throughout the deck, and most of the lower cards will be grouped together as well. It is far from random. And if you do it an 8th time, the original position is far from achieved, but while you do get a little more randomness than after the 7th shuffle, there is a huge pattern of 4 or 5 black cards and then 4 or 5 red cards throughout the entire deck, and that is after 8 shuffles. (By the way the starting position we always used was ace, king and so on, starting with Spades, then Hearts, then Diamons and then Clubs.)
However, if you do perfect shuffles with several cuts of the cards (we did three cuts, randomly between each shuffle) the pattern, after 7 perfect shuffles ALMOST completely disappeared.
So I don't know where you are getting your information, but mine is from actual trial and error, done very carefully, many times.
We also did a study with perfect shuffles, but we varied the number of cards in each stack, as is what happens in real life. Sometimes it was 30 and 22 and some times 28/24, and we found after both 7 and 8 shuffles that there still was not complete randomness, and there were some patterns, but it was more random than with equal stacks.
I can't remember several other variations, but our conclusion was that the best method of shuffling is to spread the cards out on the table and mix them up randomly, then give them a couple of shuffles, then do it again.
I would be very interested to know how the card shufflers at the casino work, as it might be possible to find patterns there that might help in blackjack, but I suspect the casinos know what they are doing and have developed some method to protect themselves from this. In my opinion, it would have to be a lot of shuffles if they are "perfect" shufflers or they must have some way to randomize the shuffles themselves. One thing I can assure you, cutting between shuffles does increase randomization and I can prove that with only one or two shuffles.

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