Mixing business cards is harder than you think

This is my summary of a project by Jiangzhen Yu in Fall 2014 -- the link goes to her write-up.


There is a mathematical theory surrounding shuffling for playing cards. That is, there is a reasonably realistic model for the usual "riffle shuffle" method, and a mathematical theory of how many shuffles are needed to make the deck approximately "random" in the usual sense of uniformly random. Here is a brief account. The conclusion has entered popular science under names like "7 shuffles are enough". The math also says that for a hypothetical deck with a large number n of cards, the number of shuffles required would be around (3/2) \log_2 n.

In contrast there is apparently no corresponding quantitative math theory for other kinds of physical mixing. Of course the machines used in modern state lotteries are tested in advance, and then in actual use, by statisticians to check that the draws are not detectably non-random but that is different from having a math theory that tells you in advance how long you should continue the mixing. A well known case of insufficient mixing was the 1969 draft lottery whose method was described as

The days of the year (including February 29) were represented by the numbers 1 through 366 written on slips of paper. The slips were placed in separate plastic capsules that were mixed in a shoebox and then dumped into a deep glass jar. Capsules were drawn from the jar one at a time.

Again in contrast to playing cards, other types of mixing involve objects which do not have standard sizes and shapes, and the details of the physical mixing procedure are difficult to describe mathematically.

Mixing business cards

Restaurants sometimes provide a large bowl into which customers may put their business card, and advertize a monthly drawing for a free lunch. We guess that no great thought is put into deciding exactly how a "random" card is drawn. But the fact that business cards are a standard size prompted us to choose them for the experiment.

As described in the write-up (link at top of this page) the experiment consisted of vigorously shaking a box of business cards -- 100, and then 200 -- a varying number of times -- from 10 to 80 times. Finally cards are draw out in top-to-bottom order and compared with the initial top-to-bottom order. One can calculate various standard "statistics", in the technical meaning of numerical measures of deviation from uniformity.

The conclusion is that the main detectable aspect of non-(uniform)-randomness is that cards which are initially adjacent tend to remain adjacent after this mixing process -- that is, a greater tendency than if completely randomized, even after 80 vigorous shakes. But other aspects of the final ordering do resemble the uniform random ordering.

Here are two possible explanations. One possible explanation is that each shake independently has the effect of shuffling large blocks of cards, but the blocks stay together internally in the same order. To be (very roughly) consistent with our data we would need the probability of adjacent cards being split into different block to be around 3%, that is for the blocks to have average size 30. An alternative explanation is that there is some physical inhomogeneity effect -- that when the cards are initially placed in the box, some adjacent pairs are "almost stuck together" in the sense of having a strong tendency to stay together which persists throughout the whole process.