Introduction

Sometimes we represent a card as a string rather than an ordered pair (for example \(\queen \, \heartsuit\) for the queen of hearts).

Contract bridge is a complex and interesting card game. We give just a very brief sketch; visit the article on contract bridge for more information. North and South are partners, as are East and West. The partners sit opposite of each other. After the cards are dealt, the players bid in turn in an auction. The winning player then has a contract to take a specified number of tricks. The winning player's partner is the dummy and lays down her hand. The winning player plays both her hand and the dummy's.

In this section our primary interest is in a couple of measures of the strength of a hand of bridge. We will use the term distribution in two ways. Of course, we know about the probability distribution of a random variable. But we will also refer to the distribution of a bridge hand by denomination or suit. Combining the two terms, we will study the probability distribution of various distributions of the bridge hand. Many of our results are given in tabular form. Longer tables are truncated and can be scrolled. All of the tables can be sorted according to a variable by clicking on the variable name in the table header. The hypergeometric distribution and the multivariate hypergeometric distribution play fundamental roles.

High-Card Distribution

When playing a trick, the players must follow suit, if they can, in which case the trick is won by the highest card played. So the four highest denominations (ace, king, queen, and jack) are particularly important, and are referred to as high cards or honor cards.

The following result gives the means, variances, covariances, and correlations.

So the mean of the vector \((A, K, Q, J, L)\) is \((1, 1, 1, 1, 9)\).

So an ace is worth 4 high-card points, a king 3, a queen 2, and a jack 1. The cards of denomination 2–10 are not awarded points. Our primary interest is in the distribution of \(V\) and fortunately we can find this from the distribution of the high-card counts. Table below gives the density function in in tabular form. The data are originally sorted in a lexicographic way in terms of the high cards. If you sort by some other variable, you can return to the default sorting by clicking on the index variable in the first column. Note that there are 610 points in the support set of the distribution.

Sorting table by hands or probability, note that mode of the probability distribution in is \((1, 1, 1, 1, 9)\), the same as the mean vector in , corresponding to the \(24\,100\,679\,680\) hands with with one ace, one king, one, queen, one jack, and 9 non-honor cards. From the distribution in we can obtain the distribution of the high card value \(V\). The computations are tedious and best done by computer. I know of no simple formula for the density function of \(V\), but table gives the results in tablular form.

Note that \(V\) takes values from 0 to 37. The distribution is unimodal with mode 10, but highly skewed to the right. The following result give the quartiles of \(V\).

As an indication of just how skewed the distribution is, the quantile of order 0.999 is 24.

Suit Distributions

In addition to the distribution of a bridge hand by denomination, the distribution by suits is also important. Here are some basic defintions.

Voids, singletons, and doubletons are important when the contract is in a suit (as opposed to no-trump) because the player may have an opportunity to take a trick in the sparse suit by trumping. On the other hand, the honor card value should be modified if there are sparse suits. For example, a queen loses much of her high card value if she's the only card in the suit.

Note that the random vector \((S, H, D, C)\) is exchangeable since any permutation of the coordinates has the same multivariate hypergeometric distribution. In particular, the variables are identically distributed.

The next definition gives a numerical measure of the strength of the hand based on the sparse suits.

In table below we list the 39 distinct suit distribution patterns (the number of cards in the suits, without specifying the acutal suits). For each pattern, the common suit value of the hands is given, as well as the number of hands and the probability. Because of the exchangeable property, the probability density function in could be derived from this table.

Note that the \([2, 3, 4, 4]\) pattern is the mode of the probability distribution and this pattern has one doubleton. The 12 suit distributions that have this pattern are the modes of the hypergeometric distribution in . Table below gives the joint distribution of the number of voids, the number of singletons, and the number of doubletons in a random bridge hand.

There are 20 points in the support set of this distribution. The mode of the distribution is \((0, 0, 1)\), corresponding to hands with one doubleton and no voids or singletons. The approximate means, variances, covariances and correlations are given next.

Of course, distinct variables are negatively correlated: more of one type of sparse suit suggests less of another type. The marginal distributions are given next.

Table next gives the probability distribution of the sparse suit value \(W\) of a bridge hand.

Note that the distribution is unimodal with mode 1, so once again corresponding to hands with one doubleton and no voids or singletons. The distribution is also highly skewed. The first quartile is 1 and the median and third quartile are both 2. The quantile of order 0.999 is 5. The mean is approximately 1.617 and the variance is approximately 0.912

More generally, long and short suits are important in judging the strength of a bridge hand, particularly when the contract is in a suit.

Table next gives the joint density function of \((L_0, L_1)\).

Note that \(L_0\) takes values in \(\{0, 1, 2, 3\}\) and \(L_1\) takes values in \(\{4, 5, \ldots, 13\}\) but not all pairs of values are possible. The support set has 21 elements. Sorting on hands or probability in table shows that the mode of the distribution is \((2, 5)\), corresponding to hands with a doubleton and a 5 card suit (and therefore suit patterns \([2, 2, 4, 5])\) or \([2, 3, 3, 5])\).

The marginal density functions of \((L_0, L_1)\) are given in table and table .

The mode of the distribution is 2, corresponding to hands with one or more doubletons but no voids or singletons. The mean is about 1.698 and the variance is about 0.5328.

The mode of the distribution is 5. The mean is about 4.901 and the variance is about 0.6959.

Missing Cards

One of the factors that makes bridge such an interesting card game is that the players gain lots of information about the bridge deal during the course of play. During the bidding, the players are communicating information about their hands to their partners, but also to the opponents. Once a contract is reached, each player sees half of the deck, but a slightly different half: her own hand and the dummy's. Then, after play begins, cards that are exposed as tricks are taken give yet more information. Through study and lots of play, good bridge players develop excellent memories for the information that has been provided at a given point in tme, and a good intuitive sense of when one event is more probable than another, given the information provided. One of the simplest examples involves the distribution of missing cards in a suit.

The distribution in is symmetric about \(k / 2\), that is \(\P(X = x) = \P(X = k - x)\) for \(x \in \{0, 1, \ldots, k\}\). The distribution is also unimodal, with a single mode at \(k / 2\) if \(k\) is even, and adjacent modes at \((k - 1) / 2\) and \((k + 1) / 2\) if \(k\) is odd. But sometimes we don't care precisely how many of the cards belong to North, but rather how the missing cards are split between North and South. By convention, we will denote the split as an ordered pair, with the smaller number first:

From the analysis in and the graphics in , it's clear that \(U\) has a folded hypergeometric distribution of sorts. Tables give the probability distribution of the splits for each \(k \in \{1, 2, \ldots, 13\}\). Naturally, our interest is in the most probable split. Sorting the table by hands or probability gives the following result.

So except when \(k = 2\), the even split is never the most probable, but rather the almost even split.