Against the Gods: The Remarkable Story of Risk (27 page)

APPENDIX: AN EXAMPLE OF THE
BAYESIAN SYSTEM OF STATISTICAL
INFERENCE IN ACTION

We return to the pin-manufacturing company. The company has two
factories, the older of which produces 40% of the total output. This
means that a pin picked up at random has a 40% probability of coming
from the old factory, whether it is defective or perfect; this is the prior
probability. We find that the older factory's defective rate is twice that
found in the newer factory. If a customer calls and complains about finding a defective pin, which of the two factories should the manager call?

The prior probability would suggest that the defective pin was most
likely to have come from the new plant, which produces 60% of the
total. On the other hand, that plant produces only one-third of the company's total of defective pins. When we revise the priors to reflect this
additional information, the probability that the new plant made the
defective pin turns out to be only 42.8%; there is a 57.2% probability that
the older plant is the culprit. This new estimate becomes the posterior
probability.

uring the last 27 years of his life, which ended at the age of 78
in 1855, Carl Friedrich Gauss slept only once away from his
home in Gottingen.' Indeed, he had refused professorships and
had declined honors from the most distinguished universities in Europe
because of his distaste for travel.

Like many mathematicians before and after him, Gauss also was a
childhood genius-a fact that displeased his father as much as it seems
to have pleased his mother. His father was an uncouth laborer who
despised the boy's intellectual precocity and made life as difficult as possible for him. His mother struggled to protect him and to encourage his
progress; Gauss remained deeply devoted to her for as long as she lived.

Gauss's biographers supply all the usual stories of mathematical miracles at an age when most people can barely manage to divide 24 by 12.
His memory for numbers was so enormous that he carried the logarithmic tables in his head, available on instant recall. At the age of eighteen, he made a discovery about the geometry of a seventeen-sided
polygon; nothing like this had happened in mathematics since the days
of the great Greek mathematicians 2,000 years earlier. His doctoral thesis, "A New Proof That Every Rational Integer Function of One
Variable Can Be Resolved into Real Factors of the First or Second Degree," is recognized by the cognoscenti as the fundamental theorem
of algebra. The concept was not new, but the proof was.

Gauss's fame as a mathematician made him a world-class celebrity. In 1807, as the French army was approaching Gottingen, Napoleon ordered his troops to spare the city because "the greatest mathematician of all times is living there."2 That was gracious of the Emperor, but fame is a two-sided coin. When the French, flushed with victory, decided to levy punitive fines on the Germans, they demanded 2,000 francs from Gauss. That was the equivalent of $5,000 in today's money and purchasing power-a heavy fine indeed for a university professor.*
A wealthy friend offered to help out, but Gauss rebuffed him. Before Gauss could say no a second time, the fine was paid for him by a distinguished French mathematician, Marquis Pierre Simon de Laplace (1749-1827). Laplace announced that he did this good deed because he considered Gauss, 29 years his junior, to be "the greatest mathematician in the world,"3 thereby ranking Gauss a few steps below Napoleon's appraisal. Then an anonymous German admirer sent Gauss 1,000 francs to provide partial repayment to Laplace.

Laplace was a colorful personality who deserves a brief digression here; we shall encounter him again in Chapter 12.

Gauss had been exploring some of the same areas of probability theory that had occupied Laplace's attention for many years. Like Gauss, Laplace had been a child prodigy in mathematics and had been fascinated by astronomy. But as we shall see, the resemblance ended there. Laplace's professional life spanned the French Revolution, the Napoleonic era, and the restoration of the monarchy. It was a time that required unusual footwork for anyone with ambitions to rise to high places. Laplace was indeed ambitious, had nimble footwork, and did rise to high places.'

In 1784, the King made Laplace an examiner of the Royal Artillery, a post that paid a handsome salary. But under the Republic, Laplace lost no time in proclaiming his "inextinguishable hatred to royalty."5 Almost immediately after Napoleon came to power, Laplace announced his enthusiastic support for the new leader, who gave him the portfolio of the Interior and the title of Count; having France's most respected sci entist on the staff added respectability to Napoleon's fledgling government. But Napoleon, having decided to give Laplace's job to his own
brother, fired Laplace after only six weeks, observing, "He was a worse
than mediocre administrator who searched everywhere for subtleties,
and brought into the affairs of government the spirit of the infinitely
small."6 So much for academics who approach too close to the seats of
power!

Later on, Laplace got his revenge. He had dedicated the 1812 edition of his magisterial Theorie analytique des probabilites to "Napoleon the
Great," but he deleted that dedication from the 1814 edition. Instead,
he linked the shift in the political winds to the subject matter of his
treatise: "The fall of empires which aspired to universal dominion," he
wrote, "could be predicted with very high probability by one versed in
the calculus of chance."' Louis XVIII took appropriate note when he
assumed the throne: Laplace became a Marquis.

Unlike Laplace, Gauss was reclusive and obsessively secretive. He
refrained from publishing a vast quantity of important mathematical
research-so much, in fact, that other mathematicians had to rediscover work that he had already completed. Moreover, his published
work emphasized results rather than his methodology, often obliging
mathematicians to search for the path to his conclusions. Eric Temple
Bell, one of Gauss's biographers, believes that mathematics might have
been fifty years further along if Gauss had been more forthcoming;
"Things buried for years or decades in [his] diary would have made half
a dozen great reputations had they been published promptly."'

Fame and secretiveness combined to make Gauss an incurable intellectual snob. Although his primary achievement was in the theory of
numbers, the same area that had fascinated Fermat, he had little use for
Fermat's pioneering work. He brushed off Fermat's Last Theorem,
which had stood as a fascinating challenge to mathematicians for over a
hundred years, as "An isolated proposition with very little interest for
me, because I could easily lay down a multitude of such propositions,
which one could neither prove nor dispose of."9

This was not an empty boast. In 1801, at the age of 24, Gauss had
published Disquisitiones Arithmeticae, written in elegant Latin, a trail blazing, historic work in the theory of numbers. Much of the book is
obscure to a non-mathematician, but what he wrote was beautiful
music to himself. 1° He found "a magical charm" in number theory and
enjoyed discovering and then proving the generality of relationships
such as this:

Or, in general, that the sum of the first n successive odd numbers is n2.
This would make the sum of the first 100 odd numbers, from 1 to 199,
equal to 1002, or 10,000; and the sum of the numbers from 1 to 999
would be equal to 250,000.

Gauss did deign to demonstrate that his theoretical work had important applications. In 1800, an Italian astronomer discovered a small new
planet-technically, an asteroid-that he named Ceres. A year later
Gauss set out to calculate its orbit; he had already calculated lunar tables
that enabled people to figure out the date of Easter in any year. Gauss
was motivated in large part by his desire to win a public reputation. But
he also wanted to join his distinguished mathematical ancestors-from
Ptolemy to Galileo and Newton-in research into celestial mechanics,
quite aside from wishing to outdo the astronomical work of his contemporary and benefactor, Laplace. In any event, this particular problem
was enticing in itself, given the paucity of relevant data and the speed
with which Ceres rotated around the sun.