Against the Gods: The Remarkable Story of Risk (11 page)

Hacking cites another example of the changing meaning of probability.12 Galileo, making explicit use of the word probabilitd, referred to
Copernicus's theory of the earth revolving around the sun as "improbable," because it contradicted what people could see with their own
eyes-the sun revolving around the earth. Such a theory was improbable because it did not meet with approval. Less than a century later,
using a new (but not yet the newest) meaning, the German scholar
Leibniz described the Copernican hypothesis as "incomparably the
most probable." For Leibniz, Hacking writes, "probability is determined by evidence and reason."13 In fact, the German word, wahrscheinlich, captures this sense of the concept well: it translates literally into
English as "with the appearance of truth."

Probability has always carried this double meaning, one looking
into the future, the other interpreting the past, one concerned with our opinions, the other concerned with what we actually know. The distinction will appear repeatedly throughout this book.

In the first sense, probability means the degree of belief or approvability of an opinion-the gut view of probability. Scholars use the
term "epistemological" to convey this meaning; epistemological refers
to the limits of human knowledge not fully analyzable.

This first concept of probability is much the older of the two; the
idea of measuring probability emerged much later. This older sense
developed over time from the idea of approbation: how much can we
accept of what we know? In Galileo's context, probability was how
much we could approve of what we were told. In Leibniz's more modern usage, it was how much credibility we could give the evidence.

The more recent view did not emerge until mathematicians had
developed a theoretical understanding of the frequencies of past events.
Cardano may have been the first to introduce the statistical side of the
theory of probability, but the contemporary meaning of the word during his lifetime still related only to the gut side and had no connection
with what he was trying to accomplish in the way of measurement.

Cardano had a sense that he was onto something big. He wrote in
his autobiography that Liber de Ludo Aleae was among his greatest
achievements, claiming that he had "discovered the reason for a thousand astounding facts." Note the words "reason for." The facts in the
book about the frequency of outcomes were known to any gambler;
the theory that explains such frequencies was not. In the book, Cardano
issues the theoretician's customary lament: ". . . these facts contribute a
great deal to understanding but hardly anything to practical play."

In his autobiography Cardano says that he wrote Liber de Ludo Aleae
in 1525, when he was still a young man, and rewrote it in 1565.
Despite its extraordinary originality, in many ways the book is a mess.
Cardano put it together from rough notes, and solutions to problems
that appear in one place are followed by solutions that employ entirely
different methods in another place. The lack of any systematic use of
mathematical symbols complicates matters further. The work was never
published during Cardano's lifetime but was found among his manuscripts when he died; it was first published in Basle in 1663. By that
time impressive progress in the theory of probability had been made by
others who were unaware of Cardano's pathfinding efforts.

Had a century not passed before Cardano's work became available for other mathematicians to build on, his generalizations about probabilities in gambling would have significantly accelerated the advance of mathematics and probability theory. He defined, for the first time, what is now the conventional format for expressing probability as a fraction: the number of favorable outcomes divided by the "circuit"-that is, the total number of possible outcomes. For example, we say the chance of throwing heads is 50/50, heads being one of two equally likely cases. The probability of drawing a queen from a full deck of cards is 1/13, as there are four queens in a deck of 52 cards; the chance of drawing the queen of spades, however, is 1/52, for the deck holds only one queen of spades.

Let us follow Cardano's line of reasoning as he details the probability of each throw in a game of dice.*
In the following paragraph from Chapter 15 of Liber de Ludo Aleae, "On the cast of one die," he is articulating general principles that no one had ever set forth before:

One-half the total number of faces always represents equality; thus the chances are equal that a given point will turn up in three throws, for the total circuit is completed in six, or again that one of three given points will turn up in one throw. For example, I can as easily throw one, three or five as two, four or six. The wagers there are laid in accordance with this equality if the die is honest.14

In carrying this line of argument forward, Cardano calculates the probability of throwing any of two numbers-say, either a 1 or a 2on a single throw. The answer is one chance out of three, or 33%, because the problem involves two numbers out of a "circuit" of six faces on the die. He also calculates the probability of repeating favorable throws with a single die. The probability of throwing a 1 or a 2 twice in succession is 1/9, which is the square of one chance out of three, or 1 /3 multiplied by itself The probability of throwing a 1 or a 2 three times in succession would be 1/27, or 1/3 x 1/3 x 1/3, while the probability of throwing a 1 or a 2 four times in succession would be 1/3 to the fourth power.

Cardano goes on to figure the probability of throwing a 1 or a 2
with a pair of dice, instead of with a single die. If the probability of
throwing a 1 or a 2 with a single die is one out of three, intuition
would suggest that throwing a 1 or a 2 with two dice would be twice
as great, or 67%. The correct answer is actually five out of nine, or
55.6%. When throwing two dice, there is one chance out of nine that
a 1 or a 2 will come up on both dice on the same throw, but the probability of a 1 or a 2 on either die has already been accounted for; hence,
we must deduct that one-ninth probability from the 67% that intuition
predicts. Thus, 1/3 + 1/3 - 1/9 = 5/9.

Cardano builds up to games for more dice and more wins more
times in succession. Ultimately, his research leads him to generalizations
about the laws of chance that convert experimentation into theory.

Cardano took a critical step in his analysis of what happens when
we shift from one die to two. Let us walk again through his line of reasoning, but in more detail. Although two dice will have a total of
twelve sides, Cardano does not define the probability of throwing a 1
or a 2 with two dice as being limited to only twelve possible outcomes.
He recognized that a player might, for example, throw a 3 on one die
and a 4 on the other die, but that the player could equally well throw
a 4 on the first die and a 3 on the second.

The number of possible combinations that make up the "circuit"the total number of possible outcomes-adds up to a lot more than the
total number of twelve faces found on the two dice. Cardano's recognition of the powerful role of combinations of numbers was the most
important step he took in developing the laws of probability.

The game of craps provides a useful illustration of the importance
of combinations in figuring probabilities. As Cardano demonstrated,
throwing a pair of six-sided dice will produce, not eleven (from two to
twelve), but thirty-six possible combinations, all the way from snake
eyes (two ones) to box cars (double six).

Seven, the key number in craps, is the easiest to throw. It is six
times as likely as double-one or double-six and three times as likely as
eleven, the other key number. The six different ways to arrive at seven
are 6 + 1, 5 + 2, 4 + 3, 3 + 4, 2 + 5, and 1 + 6; note that this pattern
is nothing more than the sums of each of three different combinations-5 and 2, 4 and 3, and 1 and 6. Eleven can show up only two
ways, because it is the sum of only one combination: 5 + 6 or 6 + 5. There is only one way for each of double-one and double-six to
appear. Craps enthusiasts would be wise to memorize this table:

In backgammon, another game in which the players throw two
dice, the numbers on each die may be either added together or considered separately. This means, for example, that, when two dice are
thrown, a 5 can appear in fifteen different ways:

The probability of a five-throw is 15/36, or about 42% I5

Semantics are important here. As Cardano put it, the probability of
an outcome is the ratio of favorable outcomes to the total opportunity
set. The odds on an outcome are the ratio of favorable outcomes to
unfavorable outcomes. The odds obviously depend on the probability,
but the odds are what matter when you are placing a bet.

If the probability of a five-throw in backgammon is 15 five-throws
out of every 36 throws, the odds on a five-throw are 15 to 21. If the
probability of throwing a 7 in craps is one out of six throws, the odds
on throwing a number other than 7 are 5 to 1. This means that you
should bet no more than $1 that 7 will come up on the next throw
when the other fellow bets $5 that it won't. The probability of heads
coming up on a coin toss are 50/50, or one out of two; since the odds
on heads are even, never bet more than your opponent on that game.
If the odds on a long-shot at the track are 20-to-1, the theoretical
probability of that nag's winning is one out of 21, or 4.8%, not 5%.

In reality, the odds are substantially less than 5%, because, unlike
craps, horse racing cannot take place in somebody's living room. Horse
races require a track, and the owners of the track and the state that
licenses the track all have a priority claim on the betting pool. If you
restate the odds on each horse in a race in terms of probabilities-as the
20-to-1 shot has a probability of winning of 4.8%-and add up the
probabilities, you will find that the total exceeds 100%. The difference
between that total and 100% is a measure of the amount that the owners and the state are skimming off the top.

We will never know whether Cardano wrote Liber de Ludo Aleae
as a primer on risk management for gamblers or as a theoretical work
on the laws of probability. In view of the importance of gambling in
his life, the rules of the game must have been a primary inspiration for
his work. But we cannot leave it at that. Gambling is an ideal laboratory in which to perform experiments on the quantification of risk.
Cardano's intense intellectual curiosity and the complex mathematical
principles that he had the temerity to tackle in Ars Magna suggest that
he must have been in search of more than ways to win at the gaming
tables.

Cardano begins Liber de Ludo Aleae in an experimental mode but
ends with the theoretical concept of combinations. Above its original insights into the role of probability in games of chance, and beyond the
mathematical power that Cardano brought to bear on the problems he
wanted to solve, Liber de Ludo Aleae is the first known effort to put measurement at the service of risk. It was through this process, which
Cardano carried out with such success, that risk management evolved.
Whatever his motivation, the book is a monumental achievement of
originality and mathematical daring.