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Bridge: Why Shuffle the Deck Seven Times?

For years, competitive bridge players complained that computer shuffling of cards produced goofy results. Statisticians sided with the computers

It’s not a magic number but it produces a fairer game.

Bridge is played with a standard 52-card deck of cards that are shuffled and dealt. Each of four players who sit at the table on opposing two-player teams receives 13 cards (a hand, pictured below). During the play of a hand, each “trick” consists of one player leading a card. The other players follow suit if they can. For example, if a player leads the 5 of spades, the other players must play a spade, too, if they have any. (The other four suits are hearts, diamonds, and clubs.)

So many possible hands could be dealt that, in practice, every hand is different—which makes for an endlessly challenging and entertaining game, popular worldwide. Indeed, bridge is one of the few games where computer algorithms have not yet demolished the best human players.

Originally, bridge hands were shuffled and dealt by the players themselves. During the late 1970s and into the early 1980s, serious competitions began switching to computer-generated hands. At first, players complained that the algorithms were faulty because they dealt too many wild hands with uneven distributions of cards. More often than they remembered, at least one player was dealt a void (no cards in one suit) or six or seven or more cards in the same suit.

These complaints were taken seriously because the players in competitive matches had many years’ experience to back up their claims that the computer-generated hands showed wilder distributions than the hands shuffled by bridge players.

Several mathematicians stepped forward and calculated the theoretical probabilities, comparing them to the actual distribution of computer-dealt hands. It turned out that the distribution of computer-generated hands was correct. For example, 18 percent of the time, at least one player should have a void; 50 percent of the time, at least one player should have 6 or more cards in the same suit; and a remarkable 15 percent of the time, at least one player should have 7 or more cards in the same suit. The computer-generated hands matched these expected frequencies.

The problem was not with the computer algorithms but with human perceptions and, as it turned out, with human shufflers.

There is widely accepted fallacy called the “law of small numbers,” according to which short-run outcomes should be similar to long-run average outcomes. If heads comes up half the time, on average, in coin flips, then we should expect 5 heads when a coin is flipped 10 times. (In fact, there is only a 25 percent chance of 5 heads and 5 tails in that case.)

In many situations, the average outcome is not only unlikely, but uninteresting, like the statistician who drowned crossing a river with an average depth of two feet. Or the British barrister who once told a group of lawyers that, “When I was a young man practicing at the bar, I lost a great many cases I should have won. As I got along, I won a great many cases I ought to have lost; so on the whole, justice was done.”

In bridge, it is tempting to think that players should expect an “average” hand, with four cards from one suit and three cards from each of the other suits. In fact, there is only a 10 percent chance of being dealt such a hand. Most hands are not average; they are far more interesting than the average.

Most bridge players—even the most experienced ones—do not appreciate how often randomly selected cards will show seemingly unusual patterns and their disbelief has been reinforced by years and years of inadequate shuffling.

The discrepancy between the veterans players’ perceptions and the mathematicians’ findings comes down to the way players shuffle cards before a match. When a bridge hand is played, many tricks have four cards from the same suit; four spades, for example. In addition, the same suit is often led two or three times in a row, causing 8 to 12 cards from the same suit to be bunched together. As a result, when the cards are collected at the end of a hand, many cards from the same suit are likely to be clustered together. If the deck is shuffled only two or three times in order to get on to the next hand (“Hurry up and deal!”), some of the bunched suits will survive largely intact and be dealt evenly to each player.

In an extreme case, a trick containing four spades may not be broken up by two or three shuffles, guaranteeing that when the cards are dealt, each player will get one of these four spades. That makes it impossible for any player to have a void in spades and difficult for any player to have seven or more spades. The reason wild hands are rare is that humans do not shuffle the cards enough!

Persi Diaconis, a Stanford University statistician and former professional magician, has shown—both in theory and practice—that if a deck of cards is divided into two equal halves and the cards are shuffled perfectly, alternating one card from each half, the deck returns to its original order after eight perfect shuffles. It is our imperfect shuffles that cause the deck to depart from its original order and it takes several flawed shuffles to mix the cards thoroughly. Diaconis and another statistician, Dave Bayer, showed that two, three, four, or even five imperfect human shuffles are not enough to randomize a deck of cards. Their rule of thumb is that seven shuffles are generally needed. Six shuffles are not enough and more than seven shuffles doesn’t have impact the randomness of the deck.

If we want random outcomes when we play bridge, poker, and other card games, we should slow down and shuffle seven times.


Note: If you enjoyed this piece by Gary Smith on randomness, both real and imagined, you might enjoy more thoughtful fun with statistics in competitive games:

The World Series: What the luck? Who will win the World Series? I don’t know, but I do know that baseball is the quintessential game of luck.

and

The paradox of luck and skill Why did Shane Lowry win the British Open golf championship? Because someone had to. In any competition including academic tests, athletic events, and company management where there is an element of luck that causes performances to be an imperfect measure of ability, there is an important difference between competitions among people with high ability and competitions among people of lesser ability.


Gary N. Smith

Gary N. Smith is the Fletcher Jones Professor of Economics at Pomona College. His research on financial markets statistical reasoning, and artificial intelligence, often involves stock market anomalies, statistical fallacies, and the misuse of data have been widely cited. He is the author of The AI Delusion, (Oxford, 2018)  and co-author (with Jay Cordes) of The 9 Pitfalls of Data Science (Oxford 2019).

Bridge: Why Shuffle the Deck Seven Times?