The Drunkard's Walk: How Randomness Rules Our Lives - Leonard Mlodinow (2008)
Chapter 3. Finding Your Way through a Space of Possibilities
IN THE YEARS leading up to 1576, an oddly attired old man could be found roving with a strange, irregular gait up and down the streets of Rome, shouting occasionally to no one in particular and being listened to by no one at all. He had once been celebrated throughout Europe, a famous astrologer, physician to nobles of the court, chair of medicine at the University of Pavia. He had created enduring inventions, including a forerunner of the combination lock and the universal joint, which is used in automobiles today. He had published 131 books on a wide range of topics in philosophy, medicine, mathematics, and science. In 1576, however, he was a man with a past but no future, living in obscurity and abject poverty. In the late summer of that year he sat at his desk and wrote his final words, an ode to his favorite son, his oldest, who had been executed sixteen years earlier, at age twenty-six. The old man died on September 20, a few days shy of his seventy-fifth birthday. He had outlived two of his three children; at his death his surviving son was employed by the Inquisition as a professional torturer. That plum job was a reward for having given evidence against his father.
Before his death, Gerolamo Cardano burned 170 unpublished manuscripts.1 Those sifting through his possessions found 111 that survived. One, written decades earlier and, from the looks of it, often revised, was a treatise of thirty-two short chapters. Titled The Book on Games of Chance, it was the first book ever written on the theory of randomness. People had been gambling and coping with other uncertainties for thousands of years. Can I make it across the desert before I die of thirst? Is it dangerous to remain under the cliff while the earth is shaking like this? Does that grin from the cave girl who likes to paint buffaloes on the sides of rocks mean she likes me? Yet until Cardano came along, no one had accomplished a reasoned analysis of the course that games or other uncertain processes take. Cardano’s insight into how chance works came embodied in a principle we shall call the law of the sample space. The law of the sample space represented a new idea and a new methodology and has formed the basis of the mathematical description of uncertainty in all the centuries that followed. It is a simple methodology, a laws-of-chance analog of the idea of balancing a checkbook. Yet with this simple method we gain the ability to approach many problems systematically that would otherwise prove almost hopelessly confusing. To illustrate both the use and the power of the law, we shall consider a problem that although easily stated and requiring no advanced mathematics to solve, has probably stumped more people than any other in the history of randomness.
AS NEWSPAPER COLUMNS GO, Parade magazine’s “Ask Marilyn” has to be considered a smashing success. Distributed in 350 newspapers and boasting a combined circulation of nearly 36 million, the question-and-answer column originated in 1986 and is still going strong. The questions can be as enlightening as the answers, an (unscientific) Gallup Poll of what is on Americans’ minds. For instance:
When the stock market closes at the end of the day, why does everyone stand around smiling and clapping regardless of whether the stocks are up or down?
A friend is pregnant with twins that she knows are fraternal. What are the chances that at least one of the babies is a girl?
When you drive by a dead skunk in the road, why does it take about 10 seconds before you smell it? Assume that you did not actually drive over the skunk.
Apparently Americans are a very practical people. The thing to note here is that each of the queries has a certain scientific or mathematical component to it, a characteristic of many of the questions answered in the column.
One might ask, especially if one knows a little something about mathematics and science, “Who is this guru Marilyn?” Well, Marilyn is Marilyn vos Savant, famous for being listed for years in the Guinness World Records Hall of Fame as the person with the world’s highest recorded IQ (228). She is also famous for being married to Robert Jarvik, inventor of the Jarvik artificial heart. But sometimes famous people, despite their other accomplishments, are remembered for something they wished had never happened (“I did not have sexual relations with that woman”). That may be the case for Marilyn, who is most famous for her response to the following question, which appeared in her column one Sunday in September 1990 (I have altered the wording slightly):
Suppose the contestants on a game show are given the choice of three doors: Behind one door is a car; behind the others, goats. After a contestant picks a door, the host, who knows what’s behind all the doors, opens one of the unchosen doors, which reveals a goat. He then says to the contestant, “Do you want to switch to the other unopened door?” Is it to the contestant’s advantage to make the switch?2
The question was inspired by the workings of the television game show Let’s Make a Deal, which ran from 1963 to 1976 and in several incarnations from 1980 to 1991. The show’s main draw was its handsome, amiable host, Monty Hall, and his provocatively clad assistant, Carol Merrill, Miss Azusa (California) of 1957.
It had to come as a surprise to the show’s creators that after airing 4,500 episodes in nearly twenty-seven years, it was this question of mathematical probability that would be their principal legacy. This issue has immortalized both Marilyn and Let’s Make a Deal because of the vehemence with which Marilyn vos Savant’s readers responded to the column. After all, it appears to be a pretty silly question. Two doors are available—open one and you win; open the other and you lose—so it seems self-evident that whether you change your choice or not, your chances of winning are 50/50. What could be simpler? The thing is, Marilyn said in her column that it is better to switch.
Despite the public’s much-heralded lethargy when it comes to mathematical issues, Marilyn’s readers reacted as if she’d advocated ceding California back to Mexico. Her denial of the obvious brought her an avalanche of mail, 10,000 letters by her estimate.3 If you ask the American people whether they agree that plants create the oxygen in the air, light travels faster than sound, or you cannot make radioactive milk safe by boiling it, you will get double-digit disagreement in each case (13 percent, 24 percent, and 35 percent, respectively).4 But on this issue, Americans were united: 92 percent agreed Marilyn was wrong.
Many readers seemed to feel let down. How could a person they trusted on such a broad range of issues be confused by such a simple question? Was her mistake a symbol of the woeful ignorance of the American people? Almost 1,000 PhDs wrote in, many of them math professors, who seemed to be especially irate.5 “You blew it,” wrote a mathematician from George Mason University:
Let me explain: If one door is shown to be a loser, that information changes the probability of either remaining choice—neither of which has any reason to be more likely—to 1/2. As a professional mathematician, I’m very concerned with the general public’s lack of mathematical skills. Please help by confessing your error and, in the future, being more careful.
From Dickinson State University came this: “I am in shock that after being corrected by at least three mathematicians, you still do not see your mistake.” From Georgetown: “How many irate mathematicians are needed to change your mind?” And someone from the U.S. Army Research Institute remarked, “If all those PhDs are wrong the country would be in serious trouble.” Responses continued in such great numbers and for such a long time that after devoting quite a bit of column space to the issue, Marilyn decided she would no longer address it.
The army PhD who wrote in may have been correct that if all those PhDs were wrong, it would be a sign of trouble. But Marilyn was correct. When told of this, Paul Erdös, one of the leading mathematicians of the twentieth century, said, “That’s impossible.” Then, when presented with a formal mathematical proof of the correct answer, he still didn’t believe it and grew angry. Only after a colleague arranged for a computer simulation in which Erdös watched hundreds of trials that came out 2 to 1 in favor of switching did Erdös concede he was wrong.6
How can something that seems so obvious be wrong? In the words of a Harvard professor who specializes in probability and statistics, “Our brains are just not wired to do probability problems very well.”7 The great American physicist Richard Feynman once told me never to think I understood a work in physics if all I had done was read someone else’s derivation. The only way to really understand a theory, he said, is to derive it yourself (or perhaps end up disproving it!). For those of us who aren’t Feynman, re-proving other people’s work is a good way to end up untenured and plying our math skills as a checker at Home Depot. But the Monty Hall problem is one of those that can be solved without any specialized mathematical knowledge. You don’t need calculus, geometry, algebra, or even amphetamines, which Erdös was reportedly fond of taking.8 (As legend has it, once after quitting for a month, he remarked, “Before, when I looked at a piece of blank paper my mind was filled with ideas. Now all I see is a blank piece of paper.”) All you need is a basic understanding of how probability works and the law of the sample space, that framework for analyzing chance situations that was first put on paper in the sixteenth century by Gerolamo Cardano.
GEROLAMO CARDANO was no rebel breaking forth from the intellectual milieu of sixteenth-century Europe. To Cardano a dog’s howl portended the death of a loved one, and a few ravens croaking on the roof meant a grave illness was on its way. He believed as much as anyone else in fate, in luck, and in seeing your future in the alignment of planets and stars. Still, had he played poker, he wouldn’t have been found drawing to an inside straight. For Cardano, gambling was second nature. His feeling for it was seated in his gut, not in his head, and so his understanding of the mathematical relationships among a game’s possible random outcomes transcended his belief that owing to fate, any such insight is futile. Cardano’s work also transcended the primitive state of mathematics in his day, for algebra and even arithmetic were yet in their stone age in the early sixteenth century, preceding even the invention of the equal sign.
History has much to say about Cardano, based on both his autobiography and the writings of some of his contemporaries. Some of the writings are contradictory, but one thing is certain: born in 1501, Gerolamo Cardano was not a child you’d have put your money on. His mother, Chiara, despised children, though—or perhaps because—she already had three boys. Short, stout, hot tempered, and promiscuous, she prepared a kind of sixteenth-century morning-after pill when she became pregnant with Gerolamo—a brew of wormwood, burned barleycorn, and tamarisk root. She drank it down in an attempt to abort the fetus. The brew sickened her, but the unborn Gerolamo was unfazed, perfectly content with whatever metabolites the concoction left in his mother’s bloodstream. Her other attempts met with similar failure.
Chiara and Gerolamo’s father, Fazio Cardano, were not married, but they often acted as if they were—they were known for their many loud quarrels. A month before Gerolamo’s birth, Chiara left their home in Milan to live with her sister in Pavia, twenty miles to the south. Gerolamo emerged after three days of painful labor. One look at the infant and Chiara must have thought she would be rid of him after all. He was frail, and worse, lay silent. Chiara’s midwife predicted he’d be dead within the hour. But if Chiara was thinking, good riddance, she was let down again, for the baby’s wet nurse soaked him in a bath of warm wine, and Gerolamo revived. The infant’s good health lasted only a few months, however. Then he, his nurse, and his three half brothers all came down with the plague. The Black Death, as the plague is sometimes called, is really three distinct diseases: bubonic, pneumonic, and septicemic plague. Cardano contracted bubonic, the most common, named for the buboes, the painful egg-size swellings of the lymph nodes that are one of the disease’s prominent symptoms. Life expectancy, once buboes appeared, was about a week.
The Black Death had first entered Europe through a harbor in Messina in northeastern Sicily in 1347, carried by a Genoese fleet returning from the Orient.9 The fleet was quickly quarantined, and the entire crew died aboard the ship—but the rats survived and scurried ashore, carrying both the bacteria and the fleas that would spread them. The ensuing outbreak killed half the city within two months and, eventually, between 25 percent and 50 percent of the population of Europe. Successive epidemics kept coming, tamping down the population of Europe for centuries. The year 1501 was a bad one for the plague in Italy. Gerolamo’s nurse and brothers died. The lucky baby got away with nothing but disfigurement: warts on his nose, forehead, cheeks, and chin. He was destined to live nearly seventy-five years. Along the way there was plenty of disharmony and, in his early years, a good many beatings.
Gerolamo’s father was a bit of an operator. A sometime pal of Leonardo da Vinci’s, he was by profession a geometer, never a profession that brought in much cash. Fazio often had trouble making the rent, so he started a consulting business, providing the highborn with advice on law and medicine. That enterprise eventually thrived, aided by Fazio’s claim that he was descended from a brother of a fellow named Goffredo Castiglioni of Milan, better known as Pope Celestine IV. When Gerolamo reached the age of five, his father brought him into the business—in a manner of speaking. That is, he strapped a pannier to his son’s back, stuffed it with heavy legal and medical books, and began dragging the young boy to meetings with his patrons all over town. Gerolamo would later write that “from time to time as we walked the streets my father would command me to stop while he opened a book and, using my head as a table, read some long passage, prodding me the while with his foot to keep still if I wearied of the great weight.”10
In 1516, Gerolamo decided his best opportunity lay in the field of medicine and announced that he wanted to leave his family’s home in Milan and travel back to Pavia to study there. Fazio wanted him to study law, however, because then he would become eligible for an annual stipend of 100 crowns. After a huge family brawl, Fazio relented, but the question remained: without the stipend, how would Gerolamo support himself in Pavia? He began to save the money he earned reading horoscopes and tutoring pupils in geometry, alchemy, and astronomy. Somewhere along the way he noticed he had a talent for gambling, a talent that would bring him cash much faster than any of those other means.
For anyone interested in gambling in Cardano’s day, every city was Las Vegas. On cards, dice, backgammon, even chess, wagers were made everywhere. Cardano classified these games according to two types: those that involved some strategy, or skill, and those that were governed by pure chance. In games like chess, Cardano risked being outplayed by some sixteenth-century Bobby Fischer. But when he bet on the fall of a couple of small cubes, his chances were as good as anyone else’s. And yet in those games he did have an advantage, because he had developed a better understanding of the odds of winning in various situations than any of his opponents. And so for his entrée into the betting world, Cardano played the games of pure chance. Before long he had set aside over 1,000 crowns for his education—more than a decade’s worth of the stipend his father wanted for him. In 1520 he registered as a student in Pavia. Soon after, he began to write down his theory of gambling.
LIVING WHEN HE DID, Cardano had the advantage of understanding many things that had been Greek to the Greeks, and to the Romans, for the Hindus had taken the first large steps toward employing arithmetic as a powerful tool. It was in that milieu that positional notation in base ten developed, and became standard, around A.D. 700.11 The Hindus also made great progress in the arithmetic of fractions—something crucial to the analysis of probabilities, since the chances of something occurring are always less than one. This Hindu knowledge was picked up by the Arabs and eventually brought to Europe. There the first abbreviations, p for “plus” and m for “minus,” were used in the fifteenth century. The symbols + and - were introduced around the same time by the Germans, but only to indicate excess and deficient weights of chests. It gives one a feeling for some of the challenges Cardano faced to note that the equal sign did not yet exist, to be invented in 1557 by Robert Recorde of Oxford and Cambridge, who, inspired by geometry, remarked that no things could be more nearly alike than parallel lines and hence decided that such lines should denote equality. And the symbol ×, for multiplication, attributable to an Anglican minister, didn’t arrive on the scene until the seventeenth century.
Cardano’s Book on Games of Chance covers card games, dice, backgammon, and astragali. It is not perfect. In its pages are reflected Cardano’s character, his crazy ideas, his wild temper, the passion with which he approached every undertaking—and the turbulence of his life and times. It considers only processes—such as the toss of a die or the dealing of a playing card—in which one outcome is as likely as another. And some points Cardano gets wrong. Still, The Book on Games of Chance represents a beachhead, the first success in the human quest to understand the nature of uncertainty. And Cardano’s method of attacking questions of chance is startling both in its power and in its simplicity.
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.12 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.13 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.
A related problem often taught in elementary probability courses is the two-daughter problem, which is similar to one of the questions I quoted from the “Ask Marilyn” column. Suppose a mother is carrying fraternal twins and wants to know the odds of having two girls, a boy and a girl, and so on. Then the sample space consists of all the possible lists of the sexes of the children in their birth order: (girl, girl), (girl, boy), (boy, girl), and (boy, boy). It is the same as the space for the coin-toss problem except for the way we name the points: heads becomes girl, and tails becomes boy. Mathematicians have a fancy name for the situation in which one problem is another in disguise: they call it an isomorphism. When you find an isomorphism, it often means you’ve saved yourself a lot of work. In this case it means we can figure the chances that both children will be girls in exactly the same way we figured the chances of both tosses coming up heads in the coin-toss problem. And so without even doing the analysis, we know that the answer is the same: 25 percent. We can now answer the question asked in Marilyn’s column: the chance that at least one of the babies will be a girl is the chance that both will be girls plus the chance that just one will be a girl—that is, 25 percent plus 50 percent, which is 75 percent.
In the two-daughter problem, an additional question is usually asked: What are the chances, given that one of the children is a girl, that both children will be girls? One might reason this way: since it is given that one of the children is a girl, there is only one child left to look at. The chance of that child’s being a girl is 50 percent, so the probability that both children are girls is 50 percent.
That is not correct. Why? Although the statement of the problem says that one child is a girl, it doesn’t say which one, and that changes things. If that sounds confusing, that’s okay, because it provides a good illustration of the power of Cardano’s method, which makes the reasoning clear.
The new information—one of the children is a girl—means that we are eliminating from consideration the possibility that both children are boys. And so, employing Cardano’s approach, we eliminate the possible outcome (boy, boy) from the sample space. That leaves only 3 outcomes in the sample space: (girl, boy), (boy, girl), and (girl, girl). Of these, only (girl, girl) is the favorable outcome—that is, both children are daughters—so the chances that both children are girls is 1 in 3, or 33 percent. Now we can see why it matters that the statement of the problem didn’t specify which child was a daughter. For instance, if the problem had asked for the chances of both children being girls given that the first child is a girl, then we would have eliminated both (boy, boy) and (boy, girl) from the sample space and the odds would have been 1 in 2, or 50 percent.
One has to give credit to Marilyn vos Savant, not only for attempting to raise public understanding of elementary probability but also for having the courage to continue to publish such questions even after her frustrating Monty Hall experience. We will end this discussion with another question taken from her column, this one from March 1996:
My dad heard this story on the radio. At Duke University, two students had received A’s in chemistry all semester. But on the night before the final exam, they were partying in another state and didn’t get back to Duke until it was over. Their excuse to the professor was that they had a flat tire, and they asked if they could take a make-up test. The professor agreed, wrote out a test, and sent the two to separate rooms to take it. The first question (on one side of the paper) was worth five points. Then they flipped the paper over and found the second question, worth 95 points: “which tire was it?” What was the probability that both students would say the same thing? My dad and I think it’s 1 in 16. Is that right?14
No, it is not: If the students were lying, the correct probability of their choosing the same answer is 1 in 4 (if you need help to see why, you can look at the notes at the back of this book).15 And now that we’re accustomed to decomposing a problem into lists of possibilities, we are ready to employ the law of the sample space to tackle the Monty Hall problem.
AS I SAID EARLIER, understanding the Monty Hall problem requires no mathematical training. But it does require some careful logical thought, so if you are reading this while watching Simpsons reruns, you might want to postpone one activity or the other. The good news is it goes on for only a few pages.
In the Monty Hall problem you are facing three doors: behind one door is something valuable, say a shiny red Maserati; behind the other two, an item of far less interest, say the complete works of Shakespeare in Serbian. You have chosen door 1. The sample space in this case is this list of three possible outcomes:
Maserati is behind door 1.
Maserati is behind door 2.
Maserati is behind door 3.
Each of these has a probability of 1 in 3. Since the assumption is that most people would prefer the Maserati, the first case is the winning case, and your chances of having guessed right are 1 in 3.
Now according to the problem, the next thing that happens is that the host, who knows what’s behind all the doors, opens one you did not choose, revealing one of the sets of Shakespeare. In opening this door, the host has used what he knows to avoid revealing the Maserati, so this is not a completely random process. There are two cases to consider.
One is the case in which your initial choice was correct. Let’s call that the Lucky Guess scenario. The host will now randomly open door 2 or door 3, and, if you choose to switch, instead of enjoying a fast, sexy ride, you’ll be the owner of Troilus and Cressida in the Torlakian dialect. In the Lucky Guess scenario you are better off not switching—but the probability of landing in the Lucky Guess scenario is only 1 in 3.
The other case we must consider is that in which your initial choice was wrong. We’ll call that the Wrong Guess scenario. The chances you guessed wrong are 2 out of 3, so the Wrong Guess scenario is twice as likely to occur as the Lucky Guess scenario. How does the Wrong Guess scenario differ from the Lucky Guess scenario? In the Wrong Guess scenario the Maserati is behind one of the doors you did not choose, and a copy of the Serbian Shakespeare is behind the other unchosen door. Unlike the Lucky Guess scenario, in this scenario the host does not randomly open an unchosen door. Since he does not want to reveal the Maserati, he chooses to open precisely the door that does not have the Maserati behind it. In other words, in the Wrong Guess scenario the host intervenes in what until now has been a random process. So the process is no longer random: the host uses his knowledge to bias the result, violating randomness by guaranteeing that if you switch your choice, you will get the fancy red car. Because of this intervention, if you find yourself in the Wrong Guess scenario, you will win if you switch and lose if you don’t.
To summarize: if you are in the Lucky Guess scenario (probability 1 in 3), you’ll win if you stick with your choice. If you are in the Wrong Guess scenario (probability 2 in 3), owing to the actions of the host, you will win if you switch your choice. And so your decision comes down to a guess: in which scenario do you find yourself? If you feel that ESP or fate has guided your initial choice, maybe you shouldn’t switch. But unless you can bend silver spoons into pretzels with your brain waves, the odds are 2 to 1 that you are in the Wrong Guess scenario, and so it is better to switch. Statistics from the television program bear this out: those who found themselves in the situation described in the problem and switched their choice won about twice as often as those who did not.
The Monty Hall problem is hard to grasp because unless you think about it carefully, the role of the host, like that of your mother, goes unappreciated. But the host is fixing the game. The host’s role can be made obvious if we suppose that instead of 3 doors, there were 100. You still choose door 1, but now you have a probability of 1 in 100 of being right. Meanwhile the chance of the Maserati’s being behind one of the other doors is 99 in 100. As before, the host opens all but one of the doors that you did not pick, being sure not to open the door hiding the Maserati if it is one of them. After he is done, the chances are still 1 in 100 that the Maserati was behind the door you chose and still 99 in 100 that it was behind one of the other doors. But now, thanks to the intervention of the host, there is only one door left representing all 99 of those other doors, and so the probability that the Maserati is behind that remaining door is 99 out of 100!
Had the Monty Hall problem been around in Cardano’s day, would he have been a Marilyn vos Savant or a Paul Erdös? The law of the sample space handles the problem nicely, but we have no way of knowing for sure, for the earliest known statement of the problem (under a different name) didn’t occur until 1959, in an article by Martin Gardner in Scientific American.16 Gardner called it “a wonderfully confusing little problem” and noted that “in no other branch of mathematics is it so easy for experts to blunder as in probability theory.” Of course, to a mathematician a blunder is an issue of embarrassment, but to a gambler it is an issue of livelihood. And so it is fitting that when it came to the first systematic theory of probability, it took Cardano, the gambler, to figure things out.
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 world.17
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.