Hostname: page-component-5c6d5d7d68-wp2c8 Total loading time: 0 Render date: 2024-08-07T08:28:50.356Z Has data issue: false hasContentIssue false
  • English
  • Français

An alternative approach to a problem by A. de Moivre

Published online by Cambridge University Press:  23 January 2015

Aristides V. Doumas
Aristides V. Doumas*
Affiliation:
Department of Mathematics, National Technical University of Athens, Zografou Campus, 157 80, Athens, Greece e-mail: aris.doumas@hotmail.com
Get access Rights & Permissions [Opens in a new window]

Extract

Suppose n (fair) dice each having m faces marked with numbers 1 to m, are thrown at random. The problem of determining the number of ways in which the sum of the numbers exhibited by the dice will be equal to a given number k has a very long history. In particular, the three dice problem (i.e. the case where m = 6, n = 3), goes back to the 13th century (see [1]). Later, it was Cardano and Galileo who solved it (see [2, 3]). The general case was stated without proof in A. de Moivre's first work on probability, De Mensura Sortis, (1712, p.220). De Moivre was the first who published a proof (using generating functions) in Miscellanea Analytica (1730, p. 196). Furthermore, Montmort had also the solution (via inclusion-exclusion) by 1713, and independently of De Moivre (see [4, 2]).

Type
Articles
Information
The Mathematical Gazette , Volume 97 , Issue 540 , November 2013 , pp. 446 - 454
Copyright
Copyright © The Mathematical Association 2013

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

1. Bellhouse, D. R., De Vetula: A medieval manuscript containing probability calculations, International Statistical Review 68 (2), (2000) pp. 123–136.Google Scholar
2. Hald, A., A. de Moivre: De mensura sortis or On the measurement of chance, International Statistical Review 52 (3), (1984) pp. 229–262.Google Scholar
3. Snell, L. J., Introduction to probability, Random House (1988).Google Scholar
4. Todhunter, I., A history of the mathematical theory of probability from the time of Pascal to that of Laplace, Macmillan (1865). Reprinted Chelsea (1949).Google Scholar
5. Feller, W., An introduction to probability theory and its applications, Vol. I, John Wiley & Sons (1968).Google Scholar
6. Karlin, S. and Taylor, H. M., A first course in stochastic processes (2nd edn.), Academic Press (1975).Google Scholar
7. Bhattacharya, R. N. and Waymire, E. C., Stochastic processes with applications, SIAM classics in Applied Mathematics, no. 61 (2009).Google Scholar
8. Lancaster, P., Theory of matrices, Academic Press (1969).Google Scholar
9. Davis, P. J., Circulant matrices, Wiley-Interscience (1979).Google Scholar
10. Gray, R. M., Toeplitz and circulant matrices: a review, Now Publishers (2006).Google Scholar
11. Lang, S., Algebra (3rd edn.), Springer-Verlag (2005).Google Scholar