When Leibniz Plays Dice

Abstract

This chapter recounts an introductory session on probability for 17-year-old students using an historic text and an IT simulation. Quite basic technically, the session’s prime aim was to introduce notions of probability, of expected value, and of equally likely outcomes. Leibniz’s text has the added interest of containing a classic error of reasoning, useful to flag up to the students. Working on an original text allows two more problematical aspects to be approached in a fairly natural context: on the one hand, the idea of the multiplicity of models for the same chance experiment; on the other, the question of the link between statistics and probability with an informal statement of the law of large numbers. This type of session, being out of the ordinary, seems a good context for approaching, if the teacher so wishes, real epistemological questions which often remain implicit in the normal run of the classroom situation.

Appendix

The SCILAB software is available on the INRIA site under the section “calcul”: http://www.inria.fr/valorisation/logiciels/index.fr.html.

“la fonction dicematrix renvoie une matrice ligne de s lancers d'un dé équilibré” function d = dicematrix(s) for i=1:s; d(1,i)=1+int(6*rand(1,1)); end; endfunction "La fonction Leibniz renvoie les couples (nombre de sommes de 5, nombre de sommes de 8) pour une répétition de n lancers de deux dés donnant l'un des deux résultats" function l = Leibniz(n) i=0 nb5 = 0 nb8 = 0 while i < n d = sum(dicematrix(2)) if d == 5 then nb5 = nb5 + 1 ; i = i+1; end if d == 8 then nb8 = nb8 + 1 ; i = i+1; end end l(1,1)=nb5 l(1,2)= nb8 endfunction "La fonction suivante donne les fréquences relatives (en %)" function l = relLeibniz(n) tirage = Leibniz(n) l(1,1)=100*tirage(1,1)/n l(1,2)=100*tirage(1,2)/n endfunction "La fonction suivante liste n répartitions de fréquences relatives sur des tirages de taille m" function l = Leibniztable(n,m) for i = 1:n a=relLeibniz(m) l(i,1)=a(1,1) l(i,2)=a(1,2) end endfunction