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Statistics

Published: Wednesday, October 6, 2021 - 11:03

In this year’s April issue of the *Bridge Bulletin,* the ACBL (American Contract Bridge League) unveiled a new logo as part of a rebranding campaign to promote the beloved game of bridge to a wider audience. Included in the new logo is the tagline “Dealing Infinite Possibilities.” It is this tagline that prompted me to write this article after several years of performing probability calculations applicable to bridge.

My background is engineering, probability, and statistics. I spent much of my career in the pulp and paper industry leading projects that required statistical design of experiments, statistical process control, data mining, and neural network analysis. So you can imagine that the ACBL tagline piqued my interest.

When I retired and decided to take up bridge again after a 45-year hiatus, I asked myself: “How many possible bridge hands are there?” It turns out the answer is not difficult to compute but does require some understanding of a concept with which most people are not familiar: Combinatorics. Combinatorics is a branch of mathematics concerned with how many ways a sample of X distinct objects can be selected from a population of Y distinct objects. Thus, when we are looking for the number of possible bridge hands there are, we are asking, “How many ways can I select 13 cards from a deck of 52 cards?” The mathematical designation for the calculation would be:

The combinations of 52 objects taken 13 at a time = _{52}C_{13} = 52!/((52-13)!13!)

The exclamation marks (!) in the above formula are called “factorials.” The factorial of a number is the product of all numbers from 1 up to that number. Thus, 3! = 1 x 2 x 3 = 6. Factorials get big really fast; 10! is greater than three million, and 13! is more than six billion.

Now, back to our calculation of how many possible bridge hands there are:

_{52}C_{13} = 52!/((52-13)!13!) = 635,013,559,600

Thus, each time you pick up a bridge hand, you are seeing one of more than 635 billion possibilities. But wait! What about the possible hands your partner and opponents have? Well, let’s think about that for a moment. Once you see your hand, the next person to look at their hand has a combination of 13 cards selected from the 39 cards remaining... _{39}C_{13} = 8,122,425,444. The third person has one of _{26}C_{13} = 10,400,600 possible combinations. What about the fourth person at the table? After the first three hands are designated, of course, the fourth seat has just _{13}C_{13} = 1 possible hand!

The next question that comes to my mind is: How many boards are possible? (A board is the total collection of four hands around the table that will be played and scored.) The answer to this question is relatively simple: We simply multiply the number of possibilities for each of the seats:

_{52}C_{13} x _{39}C_{13} x _{26}C_{13} x _{13}C_{13} = 53,644,737,765,488,800,000,000,000,000

That’s 53.6 octillion!

But wait! Is it possible that all possible boards have been played in the approximately 100-year history of bridge? Let’s see.

Suppose the population of the planet is eight billion. In that case, we imagine all eight billion people playing bridge at two billion tables (four people per table). Now let’s assume that each table plays a board every 7.5 minutes (about the average in duplicate bridge tournaments). Under these conditions, it would take eight billion people 382,739,282,002,631 years to play 53.6 octillion boards with no breaks in the action for sleep or food. Thus, it is possible but extremely unlikely that any board played in the past 100 years has been played more than once anywhere on the planet.

It is truly amazing that a simple deck of 52 cards can generate such a massive number of possibilities, and although the tenet of the ACBL’s tagline “Dealing Infinite Possibilities” is not met, it certainly seems like it.

When playing bridge, the first thing a player does is count his cards (13, if the deal was correctly performed) and assess the hand. The primary assessment tool is high card points (HCP); an ace = 4HCP, king = 3 HCP, queen = 2 HCP, and a jack is worth 1 HCP.

The bane of every bridge player is a hand with no card higher than a nine and zero HCP (such a hand is called a “Yarborough”). It happens, but with what frequency? The calculation is relatively simple; there are 32 cards ranging from the deuce to the nine; thus, the number of possible Yarboroughs is given by:

Possible number of Yarboroughs = _{32}C_{13} = 32!/((32-13)!13!) = 347,373,600

We divide this number by the total number of hands possible shown above and compute the probability as 0.000547, or once every 1,848 deals.

Aces, of course, are generally the most desirable cards to possess in a bridge hand. The results of computing the number of aces that are likely to be in a hand are interesting:

Zero aces: _{48}C_{13}/_{52}C_{13} = 30.4%

One Ace: (_{48}C_{12}/_{52}C_{13}) x 4 = 43.9%

Two aces: (_{48}C_{11}/_{52}C_{13}) x 6 = 21.3%

Three aces: (_{48}C_{10}/_{52}C_{13}) x 4 = 4.12%

Four aces: _{48}C_{9}/_{52}C_{13} = 0.26%

Notice that having one ace is more likely than having an “aceless hand.”

There are many, many other computations that can be made regarding bridge hands when you are retired and contemplating the universe... with nothing else to do. By the way, the odds of being dealt 13 spades is 635 billion to 1 against. It ain’t gonna happen!

## Comments

## How fun!

My partner and I dabble in bridge, and I dabble in math and statistics. These stats were fun to read. Thanks!