There is a small industry asking how many times you have to shuffle a deck of cards to ensure it is "randomized." Answering this question is essential to ensuring fair play in card games. If the deck is not randomized, some players may have an advantage.

Answers to this question build on a simple, but powerful, model of shuffling. These models treat a shuffle as inducing a probability measure on the set of permutations of the deck. A series of shuffles is thus a Markov chain. As these chains get longer -- as the deck is shuffled more -- the probability distribution on the permutations converges to the uniform distribution. The important question is: How fast do the distributions converge? In other words: How many times must we shuffle to ensure a fair deal?

The most famous answer, seven, was given by Bayer & Diaconis, "Trailing the Dove-tail Shuffle to Its Lair," Annals of Applied Probability Vo. 2, pp. 294-313, 1992. ("Most famous" = they made the NYTimes.)

Using a different measure of randomness (entropy instead of variation distance) Trefethen & Trefethen say shuffling five times will do: "How Many Shuffles to Randomize a Deck of Cards?," Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 456, pp. 2561-8 (2000).

Both papers are available on line, as are many, many more papers on the subject. There are also lots of blog posts discussing and summarizing the literature.

PS Your students will probably enjoy hearing a little about Diaconis' life path, which is unique under any metric.

There is a small industry asking how many times you have to shuffle a deck of cards to ensure it is "randomized." Answering this question is essential to ensuring fair play in card games. If the deck is not randomized, some players may have an advantage.

Answers to this question build on a simple, but powerful, model of shuffling. These models treat a shuffle as inducing a probability measure on the set of permutations of the deck. A series of shuffles is thus a Markov chain. As these chains get longer -- as the deck is shuffled more -- the probability distribution on the permutations converges to the uniform distribution. The important question is: How fast do the distributions converge? In other words: How many times must we shuffle to ensure a fair deal?

The most famous answer, seven, was given by Bayer & Diaconis, "Trailing the Dove-tail Shuffle to Its Lair," Annals of Applied Probability Vo. 2, pp. 294-313, 1992. ("Most famous answer" = they made the NYTimes.)

Using a different measure of randomness (entropy instead of variation distance) Trefethen & Trefethen say shuffling five times will do: "How Many Shuffles to Randomize a Deck of Cards?," Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 456, pp. 2561-8 (2000).

Both papers are available on line, as are many, many more papers on the subject. There are also lots of blog posts discussing and summarizing the literature.

PS Your students will probably enjoy hearing a little about Diaconis' life path, which is unique under any metric.

There is a small industry asking how many times you have to shuffle a deck of cards to ensure it is "randomized." Answering this question is essential to ensuring fair play in card games. If the deck is not randomized, some players may have an advantage.

Answers to this question build on a simple, but powerful, model of shuffling. These models treat a shuffle as inducing a probability measure on the set of permutations of the deck. A series of shuffles is thus a Markov chain. As these chains get longer -- as the deck is shuffled more -- the probability distribution on the permutations converges to the uniform distribution. The important question is: How fast do the distributions converge? In other words: How many times must we shuffle to ensure a fair game?

The most famous answer, seven, was given by Bayer & Diaconis, "Trailing the Dove-tail Shuffle to Its Lair," Annals of Applied Probability Vo. 2, pp. 294-313, 1992. ("Most famous" = they made the NYTimes.)

Using a different measure of randomness (entropy instead of variation distance) Trefethen & Trefethen say shuffling five times will do: "How Many Shuffles to Randomize a Deck of Cards?," Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 456, pp. 2561-8 (2000).

Both papers are available on line, as are many, many more papers on the subject. There are also lots of blog posts discussing and summarizing the literature.

PS Your students will probably enjoy hearing a little about Diaconis' life path, which is unique under any metric.

There is a small industry asking how many times you have to shuffle a deck of cards to ensure it is "randomized." Answering this question is essential to ensuring fair play in card games. If the deck is not randomized, some players may have an advantage.

Answers to this question build on a simple, but powerful, model of shuffling. These models treat a shuffle as inducing a probability measure on the set of permutations of the deck. A series of shuffles is thus a Markov chain. As these chains get longer -- as the deck is shuffled more -- the probability distribution on the permutations converges to the uniform distribution. The important question is: How fast do the distributions converge? In other words: How many times must we shuffle to ensure a fair deal?

The most famous answer, seven, was given by Bayer & Diaconis, "Trailing the Dove-tail Shuffle to Its Lair," Annals of Applied Probability Vo. 2, pp. 294-313, 1992. ("Most famous" = they made the NYTimes.)

Using a different measure of randomness (entropy instead of variation distance) Trefethen & Trefethen say shuffling five times will do: "How Many Shuffles to Randomize a Deck of Cards?," Proceedings: Mathematical, Physical and Engineering Sciences, Vol. 456, pp. 2561-8 (2000).

Both papers are available on line, as are many, many more papers on the subject. There are also lots of blog posts discussing and summarizing the literature.

PS Your students will probably enjoy hearing a little about Diaconis' life path, which is unique under any metric.