Against the Gods: The Remarkable Story of Risk (25 page)

Jacob does not use the expression "moral certainty" lightly. He
derives it from his definition of probability, which he draws from earlier
work by Leibniz. "Probability," he declares, "is degree of certainty and
differs from absolute certainty as the part differs from the whole."

But Jacob moves beyond Leibniz in considering what "certainty"
means. It is our individual judgments of certainty that attract Jacob's attention, and a condition of moral certainty exists when we are almost completely certain. When Leibniz introduced the concept, he had defined it as
"infinitely probable." Jacob himself is satisfied that 1000/1001 is close
enough, but he is willing to be flexible: "It would be useful if the magistrates
set up fixed limits for moral certainty."8

Jacob is triumphant. Now, he declares, we can make a prediction
about any uncertain quantity that will be just as scientific as the predictions made in games of chance. He has elevated probability from the
world of theory to the world of reality:

If, instead of the jar, for instance, we take the atmosphere or the
human body, which conceal within themselves a multitude of the
most varied processes or diseases, just as the jar conceals the pebbles,
then for these also we shall be able to determine by observation how
much more frequently one event will occur than another.9

Yet Jacob appears to have had trouble with his jar of pebbles. His
calculation that 25,550 trials would be necessary to establish moral
certainty must have struck him as an intolerably large number; the
entire population of his home town of Basel at that time was less than
25,550. We must surmise that he was unable to figure out what to do
next, for he ends his book right there. Nothing follows but a wistful
comment about the difficulty of finding real-life cases in which all the
observations meet the requirement that they be independent of one
another:

If thus all events through all eternity could be repeated, one would
find that everything in the world happens from definite causes and
according to definite rules, and that we would be forced to assume
amongst the most apparently fortuitous things a certain necessity, or,
so to say, FATE.1o

Nevertheless, Jacob's jar of pebbles deserves the immortality it has
earned. Those pebbles became the vehicle for the first attempt to measure uncertainty-indeed, to define it-and to calculate the probability
that an empirically determined number is close to a true value even when
the true value is an unknown.

Jacob Bernoulli died in 1705. His nephew Nicolaus-Nicolaus the
Slow-continued to work on Uncle Jacob's efforts to derive future probabilities form known observations even while he was inching along
toward the completion of Ars Conjectandi. Nicolaus's results were published in 1713, the same year in which Jacob's book finally appeared.

Jacob had started with the probability that the error between an
observed value and the true value would fall within some specified
bound; he then went on to calculate the number of observations
needed to raise the probability to that amount. Nicolaus tried to turn
his uncle's version of probability around. Taking the number of observations as given, he then calculated the probability that they would fall
within the specified bound. He used an example in which he assumed
that the ratio of male to female births was 18:17. With, say, a total of
14,000 births, the expected number of male births would be 7,200. He
then calculated that the odds are at least 43.58-to-1 that the actual
number of male births would fall between 7,200 + 163 and 7,200 -
163, or between 7,363 and 7,037.

In 1718, Nicolaus invited a French mathematician named Abraham
de Moivre to join him in his research, but de Moivre turned him
down: "I wish I were capable of... applying the Doctrine of Chances
to Oeconomical and Political Uses [but] I willingly resign my share of
that task to better Hands."11 Nevertheless, de Moivre's response to
Nicolaus reveals that the uses of probability and forecasting had come a
long way in just a few years.

De Moivre had been born in 1667-thirteen years after Jacob
Bernoulli-as a Protestant in a France that was increasingly hostile to
anyone who was not Catholic. 12 In 1685, when de Moivre was 18 years
old, King Louis XIV revoked the Edict of Nantes, which had been promulgated under the Protestant-born King Henri IV in 1598 to give
Protestants, known as Huguenots, equal political rights with Catholics.
After the revocation, exercise of the reformed religion was forbidden,
children had to be educated as Catholics, and emigration was prohibited.
De Moivre was imprisoned for over two years for his beliefs. Hating
France and everything to do with it, he managed to flee to London in
1688, where the Glorious Revolution had just banished the last vestiges
of official Catholicism. He never returned to his native country.

De Moivre led a gloomy, frustrating life in England. Despite many
efforts, he never managed to land a proper academic position. He supported himself by tutoring in mathematics and by acting as a consultant
to gamblers and insurance brokers on applications of probability theory. For that purpose, he maintained an informal office at Slaughter's Coffee
House in St. Martin's Lane, where he went most afternoons after his
tutoring chores were over. Although he and Newton were friends, and
although he was elected to the Royal Society when he was only thirty,
he remained a bitter, introspective, antisocial man. He died in 1754,
blind and poverty-stricken, at the age of 87.

In 1725, de Moivre had published a work titled Annuities upon Lives,
which included an analysis of Halley's tables on life and death in Breslaw.
Though the book was primarily a work in mathematics, it suggested
important questions related to the puzzles that the Bernoullis were trying
to resolve and that de Moivre would later explore in great detail.

Stephen Stigler, a historian of statistics, offers an interesting example
of the possibilities raised by de Moivre's work in annuities. Halley's
table showed that, of 346 men aged fifty in Breslaw, only 142, or 41%,
survived to age seventy. That was only a small sample. To what extent
could we use the result to generalize about the life expectancy of men
fifty years old? De Moivre could not use these numbers to determine the
probability that a man of fifty had a less than 50% chance of dying by age
seventy, but he would be able to answer this question: "If the true
chance were 1/2, what is the probability a ratio as small as 142/346 or
smaller should occur?"

De Moivre's first direct venture into the subject of probability was
a work titled De Mensura Sortis (literally, On the Measurement of Lots).
This work was first published in 1711 in an issue of Philosophical
Transactions, the publication of the Royal Society. In 1718, de Moivre
issued a greatly expanded English edition titled The Doctrine of Chances,
which he dedicated to his good friend Isaac Newton. The book was a
great success and went through two further editions in 1738 and 1756.
Newton was sufficiently impressed to tell his students on occasion, "Go
to Mr. de Moivre; he knows these things better than I do." De Mensura
Sortis is probably the first work that explicitly defines risk as chance of
loss: "The Risk of losing any sum is the reverse of Expectation; and the
true measure of it is, the product of the Sum adventured multiplied by
the Probability of the Loss."

In 1730, de Moivre finally turned to Nicolaus Bernoulli's project to
ascertain how well a sample of facts represented the true universe from
which the sample was drawn. He published his complete solution in
1733 and included it in the second and third editions of Doctrine of Chances. He begins by acknowledging that Jacob and Nicolaus Bernoulli
"have shewn very great skill .... [Y]et some things were farther
required." In particular, the approach taken by the Bernoullis appeared
"so laborious, and of so great difficulty, that few people have undertaken
the task."

The need for 25,550 trials was clearly an obstacle. Even if, as James
Newman has suggested, Jacob Bernoulli had been willing to settle for
the "immoral certainty" of an even bet-probability of 50/100-that
the result would be within 2% of the true ratio of 3:2, 8,400 drawings
would be needed. Jacob's selection of a probability of 1000/1001 is in
itself a curiosity by today's standards, when most statisticians accept
odds of 1 in 20 as sufficient evidence that a result is significant (today's
lingo for moral certainty) rather than due to mere chance.

De Moivre's advance in the resolution of these problems ranks
among the most important achievements in mathematics. Drawing on
both the calculus and on the underlying structure of Pascal's Triangle,
known as the binomial theorem, de Moivre demonstrated how a set of
random drawings, as in Jacob Bernoulli's jar experiment, would distribute themselves around their average value. For example, assume
that you drew a hundred pebbles in succession from Jacob's jar, always
returning each pebble drawn, and noted the ratio of white to black.
Then assume you made a series of successive drawings, each of a hundred balls. De Moivre would be able to tell you beforehand approximately how many of those ratios would be close to the average ratio of
the total number of drawings and how those individual ratios would
distribute themselves around the grand average.

De Moivre's distribution is known today as a normal curve, or,
because of its resemblance to a bell, as a bell curve. The distribution,
when traced out as a curve, shows the largest number of observations
clustered in the center, close to the average, or mean, of the total number of observations. The curve then slopes symmetrically downward,
with an equal number of observations on either side of the mean,
descending steeply at first and then exhibiting a flatter downward slope
at each end. In other words, observations far from the mean are less frequent than observations close to the mean.

The shape of de Moivre's curve enabled him to calculate a statistical
measure of its dispersion around the mean. This measure, now known
as the standard deviation, is critically important in judging whether a set of observations comprises a sufficiently representative sample of the
universe of which they are just a part. In a normal distribution, approximately 68% of the observations will fall within one standard deviation
of the mean of all the observations, and 95% of them will fall within
two standard deviations of the mean.

The standard deviation can tell us whether we are dealing with a case
of the head-in-the-oven-feet-in-the-refrigerator, where the average condition of this poor man is meaningless in telling us how he feels. Most of
the readings would be far from the average of how he felt around his
middle. The standard deviation can also tell us that Jacob's 25,550 draws
of pebbles would provide an extremely accurate estimate of the division
between the black and white pebbles inside the jar, because relatively few
observations would be outliers, far from the average.

De Moivre was impressed with the orderliness that made its appearance as the numbers of random and unconnected observations increased;
he ascribed that orderliness to the plans of the Almighty. It conveys the
promise that, under the right conditions, measurement can indeed conquer uncertainty and tame risk. Using italics to emphasize the significance
of what he had to say, de Moivre summarized his accomplishment:
"[A]tho' Chance produces Irregularities, still the Odds will be infinitely great, that
in process of Time, those Irregularities will bear no proportion to recurrency of that
Order which naturally results from ORIGINAL DESIGN."13