- PROBABILITY 513
1 2 3 4 5 6 7 8 9 10 11 12
108 111 115 120 126 133 33 36 37 36 33 26
Readers who enjoy playing with numbers may find some amusement here. Since
it is impossible to roll a 1 with three dice, the table value should perhaps be
interpreted as the number of ways in which 1 may appear on at least one of the
three dice. If so, then Cardan has got it wrong. One can imagine him thinking that
if a 1 appears on one of the dice, the other two may show 36 different numbers, and
since there are three dice on which the 1 may appear, the total number of ways of
rolling a 1 must be 3 · 36 or 108. That way of counting ignores the fact that in some
of these cases 1 appears on two of the dice or all three. By what is now known as
the inclusion-exclusion principle, the total should be 3 · 36 — 3 • 6 + 1 =91. But it is
difficult to say what Cardano had in mind. The number 111 given for 2 may be the
result of the same count, increased by the three ways of choosing two of the dice to
show a 1. Todhunter worked out a simple formula giving these numbers, but could
not imagine any gaming rules that would correspond to them. If indeed Cardano
made mistakes in his computations, he was not the only great mathematician to
do so.
Cardano's Liber de ludo (Book on Gambling) was published about a century
after his death. In this book Cardano introduces the idea of assigning a probability
ñ between 0 and 1 to an event whose outcome is not certain. The principal appli-
cations of this notion were in games of chance, where one might bet, for example,
that a player could roll a 6 with one die given three chances. The subject is not
developed in detail in Cardano's book, much of which is occupied by descriptions
of the actual games played. However, Cardano does state the multiplicative rule
for a run of successes in independent trials. Thus the probability of getting a six on
each of three successive rolls with one die is (g)^3. Most important, he recognized
the real-world application of what we call the law of large numbers, saying that
when the probability for an event is p, then after a large number ç of repetitions,
the number of times it will occur does not lie far from the value np. This law says
that it is not certain that the number of occurrences will be near np, but "that is
where the smart money bets."
After a bet has been made and before it is settled, a player cannot unilater-
ally withdraw from the bet and recover her or his stake. On the other hand, an
accountant computing the net worth of one of the players ought to count part of
the stake as an asset owned by that player; and perhaps the player would like the
right to sell out and leave the game. What would be a fair price to charge someone
for taking over the player's position? More generally, what happens if the game
is interrupted? How are the stakes to be divided? The principle that seemed fair
was that, regardless of the relative amount of the stake each player had bet, at each
moment in the game a player should be considered as owning the portion of the
stakes equal to that player's probability of winning at that moment. Thus, the net
worth of each player is constantly changing as the game progresses, in accordance
with what we now call conditional probability. Computing these probabilities in
games of chance usually involves the combinatorial counting techniques the reader
has no doubt encountered.
1.2. Fermat and Pascal. A French nobleman, the Chevalier de Mere, who was
fond of gambling, proposed to Pascal the problem of dividing the stakes in a game
where one player has bet that a six will appear in eight rolls of a single die, but