# Ivars Peterson's MathTrek

January 24, 2005

## Your mathematical chance of receiving such a hand in bridge is one in 635,013,559,600 deals

Robert D. Potter

From the Feb. 9, 1935, issue of Science News Letter.

Two of three times a year, one can pick up the morning newspaper and read something like this:

"PARIS, Mar. 4—Samuel Spadoff died here today from a heart attack caused, physicians state, by being dealt the perfect bridge hand—thirteen spades."

Or the story may have a dateline from Pittsburgh, Pasadena, or Podunk, as the case may be. And, to keep the records straight, the lucky bridge player does not always die.

The perfect bridge hand.

Thirteen spades—or any other suit to serve as trump—is the perfect bridge hand (always providing the opponents aren't clever enough to bid seven no-trump over the seven spades).

Are the stories in the papers about the "perfect" hands mere journalistic imagination, the work of joking friends who stacked the deck, or just plain unadulterated luck? Everyone who plays bridge, and there are something like 10 million bridge players in the United States along, has asked this question.

Eight years

All of which means that if every one of the 10 million bridge players in the U.S. dealt 20 bridge hands 365 days in the year, it would take the whole group of enthusiasts—or would they be lunatics?—just about 8 years, 8 months, and 1 week to deal enough hands to get a single one containing 13 spades.

The two leap years coming that time, bringing two extra days, will add a mere drop in the bucket—about 40 million deals—but perhaps it may be best to provide 2 days of rest in 8 years for the "bridgers."

No one, not even the bridge "experts," averages 20 deals a night for a year straight. That's why mathematicians and even the laymen, who admit they know little about it, scoff at the frequency with which perfect trump hand stories appear in the newspapers. The world would have to be the maniac-like place pictured to obtain the results by shear probability alone.

Go ahead and scoff. Charge off 80 or even 90 percent to hoaxes on the part of newspapermen portrayed not as they are, but after the fashion of Hollywood; anything for a laugh. Even the 10 or 20 percent left is much too high.

Practical joker?

After questioning the veracity of the newspaper story, consider the chance of the deck being stacked while the potential victim was out of the room. Suppose that once out of a million deals in the work of bridge some practical joker does stack the deck. What then is the chance that an all-spade hand will be dealt by the normal course of play?

Without plodding through the calculations, the chance is still only one in 159,000.

Lest one grow dizzy worrying about chances of a million or billion to one, it may be best to explain that the figures are all right mathematically, but conditions necessary for perfect calculations by the laws of probability are not completely satisfied in a bridge game as most people play it.

Honestly "stacked"

Why won't straight mathematical probability apply in a game of bridge? Just because most people are too lazy or haven't time and patience enough to break up the pattern of the cards by shuffling thoroughly after each completed deal.

The scoffer's suspicions that some of the "perfect" bridge hands are the result of stacked cards may be right, but not in the way they suspect. Cards have a way of stacking themselves by building up patterns through the necessity of following suit in the course of play.

You lead a space on the first round. Everyone follows suit. Thus, four spaces are grouped together. You believe spades will "go round" again so you lead another. They do. That makes eight spades in the group.

If the shuffling on the next hand is lazy, or distracted by postmortems or gossip, one may build up a deck of cards which is partially stacked.

Breaking up the card pattern is real work. In more than half of the 13 tricks in an average hand of bridge, all players will follow suit. If such a pack were dealt without shuffling, each set of four cards would go out, one to each player, and on the next deal, there would be still more tricks in which everyone followed suit. If this were repeated, eventually all four players would be getting hands containing 4-3-3-3 cards, by suits, or similar balanced patterns.

For what few "non-bridgers" there are left in the country, it may be explained that nothing helps more to make bridge monotonous as a game than these balanced hands. There are no thrills of slams and high scores when the cards start running that way.

Relief by shuffling

Just to get away from this monotony, the pack of cards is shuffled after all the 13 cards in each hand are played. Only when one has a perfect random distribution of the cards after the shuffle will the factors underlying the mathematical laws of chance enter with fair certainty.

BUILDING GOOD HANDS. It is only by thorough shuffling that the combinations of the play are broken up and the dull hands of 4-3-3-3 cards, by suits, are prevented.
\

\Just as the cards will try to work themselves into the balanced pattern, so too will they build up on rare occasions into more thrilling hands where two and more people hold seven, eight, or nine cards of a suit. Then the packets of four cards on each trick will often contain only two cards of the suit led. And the chances are better that every fourth card will be of the same suit. This means, of course, that on the next deal, chances are better that one player may receive the perfect hand.

Without trying to teach the game of bridge, the matter can be summed up by saying that mere talk of probabilities is one thing, but the game of bridge in practice is something else again.

Remember combinations

L.F. Woodruff of the electrical engineering department of the Massachusetts Institute of Technology, who has gained an additional reputation for his avocation of bridge and card probabilities, comments on how the master bridge players use their knowledge of chance and card patterns in their methods of play.

Writing in a recent issue of the Technology Review, Woodruff says, "It is almost inconceivable to the average player the number of combinations which some of the leading experts carry in their minds during match play. Not only are the probabilities considered, but at least one player of the writer's acquaintance actually makes use of the combinations that existed in the deck before the shuffle in deciding the play of the hand, on the theory that the ordinary imperfect shuffle fails to break up completely the groupings from the preceding hand."

Most people don't do it, but to start breaking up previous card patterns, four hand shuffles on each deal are necessary. Because of the cyclic nature of the average shuffle and the failure to interleave all the cards properly, an approximate approach to true randomness is hard to attain, Woodruff indicates.

For players who don't like the evenly distributed hands containing 4-3-3-3 cards in the respective suits, various dealing devices have been introduced which, by a scattered type of dealing, actually shuffle and deal the cards in one operation.

Avoids "one around"

Such "shuffle dealing" may give two cards in a row to North, then three to East, one to West, and so on, until each hand finally has its allotted 13 cards. The main idea of all these devices is to get away from the old cyclic method of dealing in rotation from left to right around the bridge table.

Woodruff, who in his spare moments devised a card "shuffle-dealing" device, has made one of the few tests to determine how the trick-taking value of hands, as measured by their long and short suit values, is affected by the number of hand shuffles after each deal.

"A hundred hands of bridge were dealt," Woodruff explains, "after each of the several procedures for shuffling the cards. Before each shuffling, the cards were grouped into tricks of four cards of the same unit. First, the cards were given one manual shuffling before dealing, for a hundred hands.

"Then 100 hands were dealt which had been shuffled manually twice; next three times, and then four times."

Increased trick value

Using a popular system of computing the trick value of the hand in long and short suits, each additional deal produced more long and more short suits and increased the trick value in this way:

• One shuffle averaged a value of 2.02 tricks per hand.
• Two shuffles averaged a value of 2.25 tricks per hand.
• Three shuffles averaged a value of 2.47 tricks per hand.
• Four shuffles averaged a value of 2.63 tricks per hand.

Completely random distributions of the cards, produced theoretically only by an infinite number of shuffles, yields—it can be calculated—an average value of 2.64 tricks a hand.

Similar studies of trick value in long and short suits for a card-dealing machine shows that an average value of 2.84 tricks for each hand could be obtained. This came about because the greatest number of cards by suits in a bridge hand, dealt for even the ideal case of perfect randomness, is of three cards each. As such, they have no trick-taking value based on the long and short suit designations.

With the card-dealing apparatus, four-card instead of three-card suits appeared with the greatest frequency, and there was an accompanying increase in two-card and one-card short suits. Either length or shortness in a suit is necessary to obtain the trick values being considered in Woodruff's research.

Moral: If you can't afford a card-dealing machine, always shuffle the pack four times.

Editor's note: Recent research has shown that even four shuffles are not enough to randomize a deck of cards. It takes at least seven shuffles to have some assurance that patterns have been broken up. See Disorder in the Deck.

References:

Peterson, I. 2002. Card shuffling shenanigans. MAA Online (Nov. 18).

______. 2000. Disorder in the deck. MAA Online (Oct. 16).

Potter, R.D. 1935. Thirteen spades. Science News Letter 27(Feb. 9):90-92.

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A collection of Ivars Peterson's early MathTrek articles, updated and illustrated, is now available as the Mathematical Association of America (MAA) book Mathematical Treks: From Surreal Numbers to Magic Circles. See http://www.maa.org/pubs/books/mtr.html.