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Diaconis on Randomness

Posted By: Timothy Chow
Date: Thursday, 23 May 2013, at 1:18 a.m.

In Response To: Diaconis on Randomness (Phil Simborg)

Phil wrote:

A very prominent expert stated that if you shuffle them well 7 times you are "assured randomness" and that there is not much point shuffling them any more than that.

The originator of this statement was, in fact, Diaconis, the man whose video bored you to tears.

I was certainly convinced that the 7-shuffle rule is false.

It's certainly false without further assumptions. What is certainly true is that under a certain model of how people shuffle, and according to a certain measure of randomness, there is a big improvement in the randomness at the 7th shuffle, and further shuffles give you a little more randomness but with diminishing returns. But as Doug Zare pointed out, it's an open question how accurate the model is. Diaconis is no fool and he did do an empirical study to check whether the model was reasonable, and found that it was pretty good. Of course, different people have different shuffling techniques, and so one must be cautious of over-generalizing.

By the way, often overlooked is the fact that one's definition of randomness can also affect the answer.

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. And if you do it an 8th time, the original position is far from achieved

There is a subtle ambiguity here in your definition of a perfect shuffle. There are two ways to perfectly interleave two stacks of 26 cards. One way leaves the original top card on top (call this an "out shuffle") and the other way doesn't (call this an "in shuffle"). If you do eight perfect out shuffles in a row then the original order will be perfectly restored.

Since you got something different on your eighth shuffle, some (or all) of your perfect shuffles must have been in shuffles. That's fine, of course, but I just wanted to point out the ambiguity, to forestall people arguing with each other when both parties are perfectly correct.

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.

You must have missed this in the video because Diaconis addressed it briefly. He was hired to analyze a particular brand of mechanical shuffler and found that it was not properly randomizing the cards. I've heard that the machines were redesigned to eliminate the flaw, but who knows if the current machines have other flaws. As usual, Wikipedia gives some useful information about how shuffling machines work.

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