Gilles Cohen was an extraordinary popularizer of mathematics. But he was far more than that. Mathematics was in his DNA. He had an instinctive feel for it. He lived and breathed it every day. Did you know that before becoming the communications expert we knew, he was also an international bridge champion? He once confided to me that he had even made a living from bridge in his youth, organizing countless rubbers in a sumptuous apartment rented for that very purpose. But what can bridge possibly have to do with mathematics? The game is a realm of randomness and therefore of chance, a fickle goddess. Yet this randomness can be tamed and mastered, mitigating the effects of poor deals while making the most of exceptional hands. And nothing does that better than statistical analysis.
Bridge is played by two partnerships sitting around a table at the four compass points, with partners opposite each other. North–South plays against East–West. The fifty-two cards in a standard deck are dealt into four hands of thirteen cards. Play unfolds in several stages, beginning with an auction. Each player must decide how many tricks their partnership will commit to taking, without knowing the other players' cards. This is a decision-making system in which information is incomplete, but not nonexistent: the bids must convey the strength of the cards held by the player making them.
At the start of the game, each player knows only their own cards. Nevertheless, the first player to bid must make a bid stating the number of tricks their partnership undertakes to take, while also choosing a suit (spades, hearts, diamonds or clubs, rather than "red" or "black") as trump. This suit plays a special role throughout the game, allowing a player, under certain conditions, to trump an opponent's card and thereby win the trick. By convention, since there are thirteen tricks in all, bids state only the number of tricks beyond six. Thus, a bid of "four diamonds" means that the declaring side must take at least ten tricks, with diamonds as trump. It is also possible to play no-trump. If players consider their hand too weak to support a bid, they may choose to "pass." Those who follow may either pass or outbid the latest proposed contract. As in almost every game, all the cards are ranked. By convention, the ace ranks above the king, which in turn ranks above the queen (where will male chauvinism rear its head next?); then come the jack, the ten and so on, with the two as the lowest card. The suits are ranked as well. Spades rank highest, followed by hearts, diamonds and clubs. This hierarchy is not without consequence. Empirical observations (see sidebar) show that most games are played at no-trump (27%), or with spades (28%) or hearts (24%) as trump, while only 11% are played with diamonds as trump and 10% with clubs.
How can players make their opening bids as rationally as possible? The number of different hands is enormous: it is the number of ways of drawing thirteen cards from fifty-two without replacement, amounting to more than 635 billion possibilities (namely the combination (52!)/(39! × 13!)). There is only one solution: statistics. To simplify the decision-making process, conventional values are assigned to the highest-ranking cards, known as honor cards. Several point scales have been tested empirically. The most widely used point count is the high-card point count (HCP), based on "high-card points": four points for an ace, three for a king, two for a queen and one for a jack. One might wonder why this system proved preferable to the others. Once again, statistics drawn from tens of thousands of games provided the answer (see sidebar).
Consider a hand with twenty high-card points in a contract with a trump suit. The observed mean number of tricks is 7.4, with a variance of 1.27 and thus a standard deviation of 1.127, giving a 95% confidence interval for the number of tricks of [5.2, 9.6]. The observed frequencies for the number of tricks are given in the following table:
The high-card point count takes eight possible values, ranging from 0 (no high cards) to 37 (four aces, four kings, four queens and one jack); such a hand has virtually zero probability and would make the game deterministic. This is a far cry from the more than 600 billion possibilities mentioned above.
Once a final contract has been bid and the next three players have passed, play itself begins. After the opening lead, the declarer's partner lays their cards face up for everyone to see and becomes the dummy, with the declarer playing those cards on their behalf. This practice changes the information available to each player and determines the strategies to adopt.
Statistics can help, but they do not determine everything. As the mathematician Bernard Charles and computer scientist Jérôme Gigault, both bridge players and authors of *Statistics and Bridge: Evaluating Hands (2006), point out: "It is important to note that point counts, like probabilities concerning the distribution of the cards, are valid only immediately after the cards are dealt. Point counts, like probabilities concerning the distribution of the cards, enable bridge players to get off to the best possible start. But from the very first bid, each player must reassess their hand, and that is above all a matter of judgment. It is fortunate for the appeal of bridge that this is so."*
