19 Dec Born in 1501, Gerolamo Cardano was not a child you’d have put your money on (LEONARD MLODINOW) | Part B’
Not all the chapters of Cardano’s book treat technical issues. For instance, chapter 26 is titled “Do
Those Who Teach Well Also Play Well?” (he concludes, “It seems to be a different thing to know and
to execute”). Chapter 29 is called “On the Character of Players” (“There are some who with many
words drive both themselves and others from their proper senses”). These seem more “Dear Abby”
than “Ask Marilyn.” But then there is chapter 14, “On Combined Points” (on possibilities). There
Cardano states what he calls “a general rule”—our law of the sample space.
The term sample space refers to the idea that the possible outcomes of a random process can be
thought of as the points in a space. In simple cases the space might consist of just a few points, but in more complex situations it can be a continuum, just like the space we live in. Cardano didn’t call it a space, however: the notion that a set of numbers could form a space was a century off, awaiting the genius of Descartes, his invention of coordinates, and his unification of algebra and geometry.
In modern language, Cardano’s rule reads like this: Suppose a random process has many equally
likely outcomes, some favorable (that is, winning), some unfavorable (losing). Then the probability
of obtaining a favorable outcome is equal to the proportion of outcomes that are favorable. The
set of all possible outcomes is called the sample space. In other words, if a die can land on any of
six sides, those six outcomes form the sample space, and if you place a bet on, say, two of them, your chances of winning are 2 in 6.
A word on the assumption that all the outcomes are equally likely. Obviously that’s not always
true. The sample space for observing Oprah Winfrey’s adult weight runs (historically) from 145
pounds to 237 pounds, and over time not all weight intervals have proved equally likely. The complication that different possibilities have different probabilities can be accounted for by associating the proper odds with each possible outcome—that is, by careful accounting. But for now
we’ll look at examples in which all outcomes are equally probable, like those Cardano analyzed.
The potency of Cardano’s rule goes hand in hand with certain subtleties. One lies in the meaning of
the term outcomes. As late as the eighteenth century the famous French mathematician Jean Le Rond d’Alembert, author of several works on probability, misused the concept when he analyzed the toss of two coins. The number of heads that turns up in those two tosses can be 0, 1, or 2. Since there are three outcomes, Alembert reasoned, the chances of each must be 1 in 3. But Alembert was mistaken.
One of the greatest deficiencies of Cardano’s work was that he made no systematic analysis of the
different ways in which a series of events, such as coin tosses, can turn out. As we shall see in the
next chapter, no one did that until the following century. Still, a series of two coin tosses is simple
enough that Cardano’s methods are easily applied. The key is to realize that the possible outcomes of coin flipping are the data describing how the two coins land, not the total number of heads calculated from that data, as in Alembert’s analysis. In other words, we should not consider 0, 1, or 2 heads as the possible outcomes but rather the sequences (heads, heads), (heads, tails), (tails, heads), and (tails, tails). These are the 4 possibilities that make up the sample space.
The next step, according to Cardano, is to sort through the outcomes, cataloguing the number of
heads we can harvest from each. Only 1 of the 4 outcomes—(heads, heads)—yields 2 heads. Similarly, only (tails, tails) yields 0 heads. But if we desire 1 head, then 2 of the outcomes are favorable: (heads, tails) and (tails, heads). And so Cardano’s method shows that Alembert was
wrong: the chances are 25 percent for 0 or 2 heads but 50 percent for 1 head. Had Cardano laid his
cash on 1 head at 3 to 1, he would have lost only half the time but tripled his money the other half, a
great opportunity for a sixteenth-century kid trying to save up money for college—and still a great
opportunity today if you can find anyone offering it.
One day while Cardano was in his teens, one of his friends died suddenly. After a few months,
Cardano noticed, his friend’s name was no longer mentioned by anyone. This saddened him and left a deep impression. How does one overcome the fact that life is transitory? He decided that the only
way was to leave something behind—heirs or lasting works of some kind or both. In his autobiography, Cardano describes developing “an unshakable ambition” to leave his mark on the
After obtaining his medical degree, Cardano returned to Milan, seeking employment. While in college he had written a paper, “On the Differing Opinions of Physicians,” that essentially called the
medical establishment a bunch of quacks. The Milan College of Physicians now returned the favor,
refusing to admit him. That meant he could not practice in Milan. And so, using money he had saved
from his tutoring and gambling, Cardano bought a tiny house to the east, in the town of Piove di
Sacco. He expected to do good business there because disease was rife in the town and it had no
physician. But his market research had a fatal flaw: the town had no doctor because the populace
preferred to be treated by sorcerers and priests. After years of intense work and study, Cardano found himself with little income but a lot of spare time on his hands. It proved a lucky break, for he seized the opportunity and began to write books. One of them was The Book on Games of Chance.
In 1532, after five years in Sacco, Cardano moved back to Milan, hoping to have his work published and once again applying for membership in the College of Physicians. On both fronts he was roundly rejected. “In those days,” he wrote, “I was sickened so to the heart that I would visit diviners and wizards so that some solution might be found to my manifold troubles.”18 One wizard suggested he shield himself from moon rays. Another that, on waking, he sneeze three times and knock on wood. Cardano followed all their prescriptions, but none changed his bad fortune. And so, hooded, he took to sneaking from building to building at night, surreptitiously treating patients who either couldn’t afford the fees of sanctioned doctors or else didn’t improve in their care. To supplement the income he earned from that endeavor, he wrote in his autobiography, he was “forced to the dice again so that I could support my wife; and here my knowledge defeated fortune, and we were able to buy food and live, though our lodgings were desolate.”19 As for The Book on Games of Chance, though he would revise and improve the manuscript repeatedly in the years to come, he never again sought to have it published, perhaps because he realized it wasn’t a good idea to teach anyone to gamble as well as he could.
Cardano eventually achieved his goals in life, obtaining both heirs and fame—and a good deal of
fortune to boot. The fortune began to accrue when he published a book based on his old college
paper, altering the title from the somewhat academic “On the Differing Opinions of Physicians” to the
zinger On the Bad Practice of Medicine in Common Use. The book was a hit. And then, when one of
his secret patients, a well-known prior of the Augustinian order of friars, suddenly (and in all likelihood by chance) improved and attributed his recovery to Cardano’s care, Cardano’s fame as a
physician took off on an upward spiral that reached such heights the College of Physicians felt
compelled not only to grant him membership but also to make him its rector. Meanwhile he was
publishing more books, and they did well, especially one for the general public called The Practice
of Arithmetic. A few years later he published a more technical book, called the Ars magna, or The
Great Art, a treatise on algebra in which he gave the first clear picture of negative numbers and a
famous analysis of certain algebraic equations. When he reached his early fifties, in the mid-1550s, Cardano was at his peak, chairman of medicine at the University of Pavia and a wealthy man.
His good fortune didn’t last. To a large extent what brought Cardano down was the other part of his
legacy—his children. When she was sixteen, his daughter Chiara (named after his mother) seduced
his older son, Giovanni, and become pregnant. She had a successful abortion, but it left her infertile. That suited her just fine, for she was boldly promiscuous, even after her marriage, and contracted
syphilis. Giovanni went on to become a doctor but was soon more famous as a petty criminal, so
famous he was blackmailed into marriage by a family of gold diggers who had proof that he had
murdered, by poison, a minor city official. Meanwhile Aldo, Cardano’s younger son who as a child
had engaged in the torture of animals, turned that passion into work as a freelance torturer for the
Inquisition. And like Giovanni, he moonlighted as a crook.
A few years after his marriage Giovanni gave one of his servants a mysterious mixture to incorporate into a cake for Giovanni’s wife. When she keeled over after enjoying her dessert, the authorities put two and two together. Despite Gerolamo’s spending a fortune on lawyers, his attempts to pull strings, and his testimony on his son’s behalf, young Giovanni was executed in prison a short while later. The drain on Cardano’s funds and reputation made him vulnerable to his old enemies. The senate in Milan expunged his name from the list of those allowed to lecture, and accusing him of sodomy and incest, had him exiled from the province. When Cardano left Milan at the end of 1563, he wrote in his autobiography, he was “reduced once more to rags, my fortune gone, my income ceased, my rents withheld, my books impounded.”20 By that time his mind was going too, and he was given to periods of incoherence. As the final blow, a self-taught mathematician named Niccolò Tartaglia, angry because in Ars magna Cardano had revealed Tartaglia’s secret method of solving certain equations, coaxed Aldo into giving evidence against his father in exchange for an official appointment as public torturer and executioner for the city of Bologna. Cardano was jailed briefly, then quietly lived out his last few years in Rome. The Book on Games of Chance was finally published in 1663, over 100 years after young Cardano had first put the words to paper. By then his methods of analysis had been reproduced and surpassed.
The Drunkard’s Walk