## Combinatorial Chance, 1

Combinatorial Chance is the title of a book by F. N. David and D. E. Barton. The title is glossed “Probability distributions of statistics of random patterns and arrangements … with applications ranging from Gambling to Non-parametric Tests”. The book was published by Charles Griffin & Co. Ltd., 42 Drury Lane, London W.C.2, in 1962, for the substantial sum of 63 shillings net.

My copy was given to me earlier this year by Hans Hockey. It is inscribed “E Whyte, Coventry 1969”. I am sure it has an interesting story.

The first author’s full name was Florence Nightingale David: she was named after the famous statistician and hospital reformer Florence Nightingale, who was a friend of her parents. The second author, David Barton, I think is the same person as a colleague of mine at Queen Mary College in the late 1980s, though both authors were at University College when the book was written.

One of the striking things about the book is its feel for history. It is dedicated to Abraham de Moivre, author of Doctrine of Chances (first edition 1718), “To his illustrious shade”. The first paragraph of the preface are worth quoting in full:

The beginnings of the calculus of probability are found in the combinatorial field, and the doctrine of combinatorial chance was firmly established by the great triumvirate Pierre-Raymond Montmort, James Bernoulli, and Abraham de Moivre. Laplace, one hundred years later, and MacMahon one hundred years after him made their significant contributions by generalising the early problems and by introducing methods of great generality and power, but often the solutions are implicit in the earlier works. It would, of course, be untrue to assume that every piece of combinatorial theory may be traced back to the Newtonian era; on the other hand it is remarkable how much is “discovered” today which can be found, if not in de Moivre’s Doctrine of Chances, in Laplace’s Théorie Analytique des Probabilités. THe increased interest in combinatorial chance since the end of the last war has brought a number of such false dawns.

A salutary reminder. I hope in a later post to say more about the contents of the book, but here I shall mainly concentrate on the first chapter, “Elementary Combinatorial Method: Illustrative Exercises”, which leans heavily on much older work, and yet probably has something new to most combinatorialists today.

For the number of combinations of a set of n things, they give the formula 2nn−1, from William Buckley in 1567. (To Buckley and his contemporaries, a single element, or no element at all, would not constitute a “combination”.) The authors say, “… a proposition usually attributed to Pascal. It is to be questioned, however, whether the priority is his, since it follows as a natural consequence of the Arithmetic Triangle, which appears in a thirteenth-century Chinese treatise, and was known in Europe by the fifteenth century.”

In Latin, because of the elaborate system of cases and declensions, word-order is not so important. Yet the authors cite Bernard Bauhusius, “some time before 1617”, who discussed the number of possible arrangements of words of the line

Tot tibi sunt dotes, Virgo, quot sidera caelo,

which are “not derogatory to the Virgin”, and came to the conclusion that there are 1,022. My Latin is not up to this: can anyone help? Puteanus pointed out that this is the number of the stars in Ptolemy’s catalogue. Of course the total number of arrangements is 8! = 40,320, so the proportion which “convey the glory of the Virgin” is about 2.5%. (The authors don’t make clear whether Bauhusius know this.)

On page 7 a problem put to Galileo by the Duke of Tuscany in about 1613 is posed and solved: “There are six 3-partitions of the number 10 and of the number 12, and yet when three six-sided dice are thrown simultaneously 10 appears more frequently than 12. What is the explanation?”

Another problem on pages 9–11 shows a curious coincidence and links up with my preceding post (this is completely by chance!)

N boxes are arranged in a line and numbered 1, 2, … N. Into the boxes n balls are dropped randomly, not more than one ball in any box. The range is the difference between the numbers of the first and last occupied box. The distribution of the range is calculated, and the results for N = 10, n = 5, are tabulated.

Then the distribution of the position of the ith ball is calculated (the balls are ordered according to the boxes containing them), and the results for N = 10, n = 5, i = 2, are tabulated. The authors make no comment on the fact that the numbers in this table are obtained from the preceding table by reflection and shift, leaving it to their intelligent readers to observe that, from their formulae, the probability that the second ball is in position r is equal to the probability that the range is N+1−r.

They do check that the probabilities in the latter case sum to 1. This involves proving the identity

$\sum_{r=1}^{N-n+i}{r-1\choose i-1}{N-r\choose n-i}={N\choose n},$

which happens to be an instance of the dual Vandermonde convolution I mentioned in my last post.