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

### Background

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.