[SamIAm]

[ QUOTE ]
7 shuffles

[/ QUOTE ]I'm really happy that people throw this answer around. I mean, it's a cool result of Persi Diaconis, but most people giving the answer don't know the question.

For instance, what does "shuffle" mean? I know it's "cut & riffle", but it certainly matters how much variance there is in your cut AND riffle. (An interesting side note is that if your shuffle has NO variance, i.e. a perfect shuffle, then 8 "shuffles" will return the deck to the original state. A perfect shuffle is cutting 26/26, and then riffling alternate L,R,L,R,... perfectly. This clearly isn't randomizing at all.)

In the Diaconis paper, I think they assume that you cut with the binomial distribution (so the chance of cutting X cards is the chance of getting X Heads in 52 coint flips) and that you riffle proportional to the numbers of cards in each side. For instance, if you have 30 in the left side and 22 in the other, a left card will fall with probability 30/52. Then you repeat. These are reasonable assumptions, but of course they aren't perfect.

A much more important question is "What does 'adequate' mean?" There's no way to get it perfectly random; there's no such thing. However, with every iteration, we get closer to random. You just have to define how close you want it to get.

The way probabilists define "distance from random" is the "total variation distance". Basicially, you look at the set of decks that have the most incorrect chance of coming-up. If there's some set of decks D with the probability of getting one of those = |D|/52!+X, and this set has the largest such "X", then we say the TotalVariation Distance is X. The Diaconis paper defined some minumum X, and found that 7 shuffles get the T.V. dist to within X. But their choice of X is pretty arbitrary. (I forget how they justified their specific choice.)

My point is, there are lots of arbitrary decisions that went into this "7" number. The assumption of shuffle mechanics, of cut mechanics, and of "adequate". It sure makes good press, though. [img]/images/graemlins/smile.gif[/img]
-Sam