Euler's Analysis of the Genoese Lottery - Conclusion

Euler wrote a total four papers on the Genoese lottery, as well as a fifth one [13] on an entirely different type of lottery, which was also proposed to Frederick II as a way to raise money, this time by a Dutchman named van Griethouse.

The second of the four Genoese lottery papers, called "On the Probability of Sequences in the Genoese Lottery" [14], was read to the Berlin Academy of Sciences in 1765 and published in the Mémoires of the academy two years later. In this memoir, Euler considers the probability that sequences, or runs of consecutive numbers, will appear among the numbers drawn in a Genoese style lottery. If, for example, the numbers 7, 8, 25, 26 and 27 are drawn, 7 and 8 constitute a sequence of two whereas 25, 26 and 27 constitute a sequence of three. It was not possible to bet on the occurrence of sequences in the Berlin lottery, so we must assume that this is simply a mathematical puzzle which Euler found appealing and worthy of his attention.

After leaving Berlin in 1766, Euler wrote two more memoirs concerning matters of probability theory arising from the Genoese lottery. Both of these papers [15, 16] concerned the number of distinct integers between 1 and n which would be drawn over the course of many repetitions of the lottery. The first of these, "On the Solution of Difficult Questions in the Calculus of Probability" was presented to the Academy of St. Petersburg on October 8, 1781 but published in 1785, two years after Euler's death. Euler begins with the observation that it would be convenient to have a special symbol for the binomial coefficient, and so he defines:

\({p \choose q}=\frac{p(p-1) \cdots (p-q+1)}{q!}\)

Actually, Euler used square brackets in place of the parentheses in the original published version of this paper, but in a variety of roughly contemporary papers, he used the above notation, and so this is what appears in a modern edition of the paper. So it was that Euler, who invented so much of the notation we still use in mathematics, pioneered one modern symbol for the binomial coefficient.

The final paper, entitled "Analysis of a Problem in the Calculus of Probability," was published posthumously in 1862. It is shorter than his other other three papers on the Genoese lottery, and reads more tersely when compared with the lucid technical prose of those three. It was probably a preliminary draft that Euler never managed to put into final form.

All in all, Euler's writings on lotteries comprise but a tiny fraction of his total mathematical output, but they stand as a testament to his love of puzzles and mathematical recreations.

Editor’s note: This article was published in April 2004.