Girolamo Cardano is a transitional person in the history of probability. Of course, every mathematician worthy of note is, in some sense, a transi- tional figure; each good mathematician corrects past errors and con- tributes something to future progress. But the statement has a special meaning for Cardano. Because of his mathematical background he was able to identify a new way of thinking about games of chance.
Cardano was sometimes able to understand and use probability in ways that sound modern. For example, he knew that the odds of throw- ing a 10 with two dice are 1/12. He finds this by counting the number of favorable outcomes. There are, he tells us, three ways of obtaining a 10 with two dice. One can roll
(5, 5), that is, a 5 on each die, or
(6, 4), that is, a 6 on the first die and a 4 on the second, or
(4, 6), a 4 on the first die and a 6 on the second.
Next notice that there are 36 different outcomes. To see why, imagine that one red and one green die are used—that way we can distinguish between them. If 1 is rolled with the red die, that 1 can be paired with any of six numbers—that is, 1, 2, 3, 4, 5, and 6—rolled on the green die. So there are six possible outcomes associated with rolling a red 1. There is, however, nothing special about the number 1. Exactly the same argument can be used for any other number that appears on the red die. Summing up all the possibilities we get 36 different possible outcomes. (See the accompanying chart.)
Divide the sum of favorable outcomes (3) by the number of possible outcomes (36) and one obtains 3/36 or 1/12. It is a simple result, but it shows that he understands the principle involved.
What is interesting about Cardano is that although he understands how to calculate the odds for certain simple outcomes, he does not quite believe in the calculation. The difficulty that he has in interpreting his calculations arises from the fact that he cannot quite jettison the very unscientific idea of luck. Here is an excerpt from a section of Liber de
Ludo Alea entitled “On Timidity in the Throw.”
For this reason it is natural to wonder why those who throw the dice timidly are defeated. Does the mind itself have a presentiment of evil? But we must free men from error; for
although this might be thought true, still we have a more mani- fest reason. For when anyone begins to succumb to adverse fortune, he is very often accustomed to throw the dice timidly; but if the adverse fortune persists, it will necessarily fall unfa- vorably. Then, since he threw it timidly, people think that it fell unfavorably for that very reason; but this is not so. It is because fortune is adverse that the die falls unfavorably, and because the die falls unfavorably he loses, and because he loses he throws the die timidly.
In Liber de Ludo Aleae we find luck and math side by side. That is part of what makes the book so interesting to a modern reader.
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"First" Die S e c o n d D i eThe table shows all 36 possible outcomes that can be obtained by rolling two dice. The three shaded squares indicate the three possible ways of rolling a 10:6 on the first die and 4 on the second, 5 on each die, and 6 on the second die and 4 on the first.
A modern reader can occasionally find it a little frustrating (or a little humorous) to read the Liber de Ludo Aleae. One begins to wonder when Cardano will get around to drawing the “obvious” conclusions. He usually does not. He points out, for example, that if one chooses any three sides of a die, then the numbers on those three sides are just as likely to show on one roll of the die as the numbers on the other three sides. From this he concludes, “I can as easily throw one, three or five as two, four or six” (ibid.). In a sense, by marking out three faces of a six-sided die as favorable and three as unfavorable he turned the problem of rolling a die into a coin-toss problem: The odds are 50/50, he tells us, that we will roll either a 1, a 3, or a 5. He was right, of course, and he did go a lit- tle beyond this simple case, but his understanding of probability, even as it relates exclusively to dice, was very limited.
Mathematically, he came very close to making deeper discover- ies, but he never quite made the necessary connections. Moreover, not every mathematically formulated remark that he wrote about dice is correct. He concludes, for example, that if one throws a die three times the chance that a given number will show at least once is 50 percent, whereas it is actually about 42 percent. To be sure, he did not get very far in his analysis, but it is important to keep in mind that he was the first to attempt to formulate probabilistic descriptions of “random phenomenon.”
We can develop a fuller appreciation of Cardano’s work if we keep in mind two additional barriers that Cardano faced in addition to the newness of the subject. First, it would have been hard for anyone to develop a more comprehensive theory of probability without a good system of algebraic notation. Without algebra it is much harder to represent one’s mathematical ideas on paper, and in Cardano’s time the algebraic notation necessary for expressing basic probability was still in the process of being developed. (Liber de Ludo Aleae is practi- cally all prose.) Second, although Cardano stood at the edge of a new way of thinking about randomness, it is clear that he could not quite let go of the old ideas. In particular, he could not lose the old pre- conceptions about the role of luck. He was very sure, for example, that the attitude of the person throwing the dice affects the outcome of the throw. (Over a century later the great mathematician Abraham
de Moivre felt it necessary to include a section in his book The
Doctrine of Chances repudiating the idea that luck is something that
can affect the outcome of a random event.) Although he could com- pute simple odds, Cardano was unwilling to let the numbers speak for themselves. Luck, he believed, still played a part.
Despite these shortcomings we find in Cardano’s writings the first evidence of someone’s attempting to develop a mathematical description of random patterns.