1.2. DISCRETE PROBABILITY DISTRIBUTIONS
31
be 0, 1, or 2, and hence we should assign equal probabilities for these three possible
outcomes.
16
Cardano chose the correct sample space for his dice problems and
calculated the correct probabilities for a variety of events.
Cardano’s mathematical work is interspersed with a lot of advice to the potential
gambler in short paragraphs, entitled, for example: “Who Should Play and When,”
“Why Gambling Was Condemned by Aristotle,” “Do Those Who Teach Also Play
Well?” and so forth. In a paragraph entitled “The Fundamental Principle of Gam
bling,” Cardano writes:
The most fundamental principle of all in gambling is simply equal con
ditions, e.g., of opponents, of bystanders, of money, of situation, of the
dice box, and of the die itself. To the extent to which you depart from
that equality, if it is in your opponent’s favor, you are a fool, and if in
your own, you are unjust.
17
Cardano did make mistakes, and if he realized it later he did not go back and
change his error. For example, for an event that is favorable in three out of four
cases, Cardano assigned the correct odds 3 : 1 that the event will occur. But then he
assigned odds by squaring these numbers (i.e., 9 : 1) for the event to happen twice in
a row. Later, by considering the case where the odds are 1 : 1, he realized that this
cannot be correct and was led to the correct result that when
f
out of
n
outcomes
are favorable, the odds for a favorable outcome twice in a row are
f
2
:
n
2

f
2
. Ore
points out that this is equivalent to the realization that if the probability that an