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1.4. Leibniz. Although Leibniz wrote a full treatise on combinatorics, which pro-
vides the mathematical apparatus for computing many probabilities in games of
chance, he did not himself gamble. But he did analyze many games of chance and
suggest modifications of them that would make them fair (zero-sum) games. Some
of his manuscripts on this topic have been analyzed by de Mora-Charles (1992).
One of the games he analyzed is known as quinquenove. This game is played be-
tween two players using a pair of dice. One of the players, called the banker, rolls
the dice, winning if the result is either a double or a total number of spots showing
equal to 3 or 11. There are thus 10 equally likely ways for the banker to win with
this roll, out of 36 equally likely outcomes. If the banker rolls a 5 or 9 (hence the
name "quinquenove"), the other player wins. The other player has eight ways of
winning of the equally likely 36 outcomes, leaving 18 ways for the game to end in
a draw. The reader will be fascinated and perhaps relieved to learn that the great
Leibniz, author of De arte combinatorial confused permutations and combinations
in his calculations for this game and got the probabilities wrong.
1.5. The Ars Conjectandi of Jakob Bernoulli. One of the classic founding
documents of probability theory was published in 1713, eight years after the death
of its author, Leibniz' disciple Jakob Bernoulli. This work, Ars conjectandi (The
Art of Prediction), moved probability theory beyond the limitations of analyzing
games of chance. It was intended by its author to apply mathematical methods to
the uncertainties of life. As he said in a letter to Leibniz, "I have now finished the
major part of the book, but it still lacks the particular examples, the principles of the
art of prediction that I teach how to apply to society, morals, and economics... ."
That was an ambitious undertaking, and Bernoulli had not quite finished the work
when he died in 1705.
Bernoulli gave a very stark picture of the gap between theory and application,
saying that only in simple games such as dice could one apply the equal-likelihood
approach of Fermat and Pascal, whereas in the cases of interest, such as human
health and longevity, no one had the power to construct a suitable model. He
recommended statistical studies as the remedy to our ignorance, saying that if 200
people out of 300 of a given age and constitution were known to have died within
10 years, it was a 2-to-l bet that any other person of that age and constitution
would die within a decade.
In this treatise Bernoulli reproduced the problems solved by Huygens and gave
his own solution of them. He considered what are now called Bernoulli trials in
his honor. These are repeated experiments in which a particular outcome either
happens (success) with probability b/a or does not happen (failure) with probability
c/a, the same probability each time the experiment is performed, each outcome
being independent of all others. (A simple nontrivial example is rolling a single
die, counting success as rolling a 5. Then the probabilities are g and |.) Since
b/a + c/a = 1, Bernoulli saw correctly that the binomial expansion, and hence
Pascal's triangle, would be useful in computing the probability of getting at least
m successes in ç trials. He gave that probability as
It was, incidentally, in this treatise, when computing the sum of the cth powers
of the first ç integers, that Bernoulli introduced what are now called the Bernoulli