Not all the chapters of Cardano\u2019s book treat technical issues. For instance, chapter 26 is titled \u201cDo
\nThose Who Teach Well Also Play Well?\u201d (he concludes, \u201cIt seems to be a different thing to know and
\nto execute\u201d). Chapter 29 is called \u201cOn the Character of Players\u201d (\u201cThere are some who with many
\nwords drive both themselves and others from their proper senses\u201d). These seem more \u201cDear Abby\u201d
\nthan \u201cAsk Marilyn.\u201d But then there is chapter 14, \u201cOn Combined Points\u201d (on possibilities). There
\nCardano states what he calls \u201ca general rule\u201d\u2014our law of the sample space.<\/p>\n
The term sample space refers to the idea that the possible outcomes of a random process can be
\nthought 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\u2019t 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.<\/p>\n
In modern language, Cardano\u2019s rule reads like this: Suppose a random process has many equally
\nlikely outcomes, some favorable (that is, winning), some unfavorable (losing). Then the probability
\nof obtaining a favorable outcome is equal to the proportion of outcomes that are favorable. The
\nset of all possible outcomes is called the sample space. In other words, if a die can land on any of
\nsix 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.<\/p>\n
A word on the assumption that all the outcomes are equally likely. Obviously that\u2019s not always
\ntrue. The sample space for observing Oprah Winfrey\u2019s adult weight runs (historically) from 145
\npounds 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\u2014that is, by careful accounting. But for now
\nwe\u2019ll look at examples in which all outcomes are equally probable, like those Cardano analyzed.<\/p>\n
The potency of Cardano\u2019s rule goes hand in hand with certain subtleties. One lies in the meaning of
\nthe term outcomes. As late as the eighteenth century the famous French mathematician Jean Le Rond d\u2019Alembert, 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.<\/p>\n
One of the greatest deficiencies of Cardano\u2019s work was that he made no systematic analysis of the
\ndifferent ways in which a series of events, such as coin tosses, can turn out. As we shall see in the
\nnext chapter, no one did that until the following century. Still, a series of two coin tosses is simple
\nenough that Cardano\u2019s 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\u2019s 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.<\/p>\n
The next step, according to Cardano, is to sort through the outcomes, cataloguing the number of
\nheads we can harvest from each. Only 1 of the 4 outcomes\u2014(heads, heads)\u2014yields 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\u2019s method shows that Alembert was
\nwrong: the chances are 25 percent for 0 or 2 heads but 50 percent for 1 head. Had Cardano laid his
\ncash on 1 head at 3 to 1, he would have lost only half the time but tripled his money the other half, a
\ngreat opportunity for a sixteenth-century kid trying to save up money for college\u2014and still a great
\nopportunity today if you can find anyone offering it.<\/p>\n
One day while Cardano was in his teens, one of his friends died suddenly. After a few months,
\nCardano noticed, his friend\u2019s 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
\nway was to leave something behind\u2014heirs or lasting works of some kind or both. In his autobiography, Cardano describes developing \u201can unshakable ambition\u201d to leave his mark on the
\nworld.<\/p>\n
After obtaining his medical degree, Cardano returned to Milan, seeking employment. While in college he had written a paper, \u201cOn the Differing Opinions of Physicians,\u201d that essentially called the
\nmedical establishment a bunch of quacks. The Milan College of Physicians now returned the favor,
\nrefusing to admit him. That meant he could not practice in Milan. And so, using money he had saved
\nfrom his tutoring and gambling, Cardano bought a tiny house to the east, in the town of Piove di
\nSacco. He expected to do good business there because disease was rife in the town and it had no
\nphysician. But his market research had a fatal flaw: the town had no doctor because the populace
\npreferred 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.<\/p>\n
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. \u201cIn those days,\u201d he wrote, \u201cI was sickened so to the heart that I would visit diviners and wizards so that some solution might be found to my manifold troubles.\u201d18 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\u2019t afford the fees of sanctioned doctors or else didn\u2019t improve in their care. To supplement the income he earned from that endeavor, he wrote in his autobiography, he was \u201cforced 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.\u201d19 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\u2019t a good idea to teach anyone to gamble as well as he could.<\/p>\n
Cardano eventually achieved his goals in life, obtaining both heirs and fame\u2014and a good deal of
\nfortune to boot. The fortune began to accrue when he published a book based on his old college
\npaper, altering the title from the somewhat academic \u201cOn the Differing Opinions of Physicians\u201d to the
\nzinger On the Bad Practice of Medicine in Common Use. The book was a hit. And then, when one of
\nhis 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\u2019s care, Cardano\u2019s fame as a
\nphysician took off on an upward spiral that reached such heights the College of Physicians felt
\ncompelled not only to grant him membership but also to make him its rector. Meanwhile he was
\npublishing more books, and they did well, especially one for the general public called The Practice
\nof Arithmetic. A few years later he published a more technical book, called the Ars magna, or The
\nGreat Art, a treatise on algebra in which he gave the first clear picture of negative numbers and a
\nfamous 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.<\/p>\n
His good fortune didn\u2019t last. To a large extent what brought Cardano down was the other part of his
\nlegacy\u2014his children. When she was sixteen, his daughter Chiara (named after his mother) seduced
\nhis 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
\nsyphilis. Giovanni went on to become a doctor but was soon more famous as a petty criminal, so
\nfamous he was blackmailed into marriage by a family of gold diggers who had proof that he had
\nmurdered, by poison, a minor city official. Meanwhile Aldo, Cardano\u2019s younger son who as a child
\nhad engaged in the torture of animals, turned that passion into work as a freelance torturer for the
\nInquisition. And like Giovanni, he moonlighted as a crook.<\/p>\n
A few years after his marriage Giovanni gave one of his servants a mysterious mixture to incorporate into a cake for Giovanni\u2019s wife. When she keeled over after enjoying her dessert, the authorities put two and two together. Despite Gerolamo\u2019s spending a fortune on lawyers, his attempts to pull strings, and his testimony on his son\u2019s behalf, young Giovanni was executed in prison a short while later. The drain on Cardano\u2019s 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 \u201creduced once more to rags, my fortune gone, my income ceased, my rents withheld, my books impounded.\u201d20 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 Niccol\u00f2 Tartaglia, angry because in Ars magna Cardano had revealed Tartaglia\u2019s 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.<\/p>\n
<\/p>\n
<\/p>\n
Part A’:\u00a0http:\/\/www.lecturesbureau.gr\/1\/born-in-1501-gerolamo-cardano-was-not-a-child-youd-have-put-your-money-on-leonard-mlodinow-part-a\/?lang=en<\/a><\/p>\n <\/p>\n <\/p>\n The Drunkard’s Walk<\/em><\/strong> Not all the chapters of Cardano\u2019s book treat technical issues. For instance, chapter 26 is titled \u201cDo Those Who Teach Well Also Play Well?\u201d (he concludes, \u201cIt seems to be a different thing to know and to execute\u201d). Chapter 29 is called \u201cOn the Character…<\/p>\n","protected":false},"author":1,"featured_media":24911,"comment_status":"open","ping_status":"open","sticky":false,"template":"","format":"standard","meta":{"_monsterinsights_skip_tracking":false,"_monsterinsights_sitenote_active":false,"_monsterinsights_sitenote_note":"","_monsterinsights_sitenote_category":0,"_jetpack_memberships_contains_paid_content":false,"footnotes":""},"categories":[88],"tags":[],"jetpack_sharing_enabled":true,"jetpack_featured_media_url":"https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?fit=900%2C609&ssl=1","rttpg_featured_image_url":{"full":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?fit=900%2C609&ssl=1",900,609,false],"landscape":["https:\/\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg",900,609,false],"portraits":["https:\/\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg",900,609,false],"thumbnail":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=150%2C150&ssl=1",150,150,true],"medium":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?fit=300%2C203&ssl=1",300,203,true],"large":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?fit=900%2C609&ssl=1",900,609,true],"1536x1536":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?fit=900%2C609&ssl=1",900,609,true],"2048x2048":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?fit=900%2C609&ssl=1",900,609,true],"portfolio-square":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=570%2C570&ssl=1",570,570,true],"portfolio-portrait":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=600%2C609&ssl=1",600,609,true],"portfolio-landscape":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=800%2C600&ssl=1",800,600,true],"menu-featured-post":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=345%2C198&ssl=1",345,198,true],"qode-carousel_slider":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=400%2C260&ssl=1",400,260,true],"portfolio_slider":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=500%2C380&ssl=1",500,380,true],"portfolio_masonry_regular":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=500%2C500&ssl=1",500,500,true],"portfolio_masonry_wide":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=900%2C500&ssl=1",900,500,true],"portfolio_masonry_tall":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=500%2C609&ssl=1",500,609,true],"portfolio_masonry_large":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=900%2C609&ssl=1",900,609,true],"portfolio_masonry_with_space":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?fit=700%2C474&ssl=1",700,474,true],"latest_post_boxes":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=539%2C303&ssl=1",539,303,true],"woocommerce_thumbnail":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=300%2C300&ssl=1",300,300,true],"woocommerce_single":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?fit=600%2C406&ssl=1",600,406,true],"woocommerce_gallery_thumbnail":["https:\/\/i0.wp.com\/www.lecturesbureau.gr\/1\/wp-content\/uploads\/2017\/12\/post-1081b.jpg?resize=100%2C100&ssl=1",100,100,true]},"rttpg_author":{"display_name":"admin","author_link":"https:\/\/www.lecturesbureau.gr\/1\/author\/admin\/"},"rttpg_comment":0,"rttpg_category":"Science<\/a>","rttpg_excerpt":"Not all the chapters of Cardano\u2019s book treat technical issues. For instance, chapter 26 is titled \u201cDo Those Who Teach Well Also Play Well?\u201d (he concludes, \u201cIt seems to be a different thing to know and to execute\u201d). Chapter 29 is called \u201cOn the Character...","_links":{"self":[{"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/posts\/24942"}],"collection":[{"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/posts"}],"about":[{"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/types\/post"}],"author":[{"embeddable":true,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/users\/1"}],"replies":[{"embeddable":true,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/comments?post=24942"}],"version-history":[{"count":1,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/posts\/24942\/revisions"}],"predecessor-version":[{"id":24943,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/posts\/24942\/revisions\/24943"}],"wp:featuredmedia":[{"embeddable":true,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/media\/24911"}],"wp:attachment":[{"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/media?parent=24942"}],"wp:term":[{"taxonomy":"category","embeddable":true,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/categories?post=24942"},{"taxonomy":"post_tag","embeddable":true,"href":"https:\/\/www.lecturesbureau.gr\/1\/wp-json\/wp\/v2\/tags?post=24942"}],"curies":[{"name":"wp","href":"https:\/\/api.w.org\/{rel}","templated":true}]}}
\nLEONARD MLODINOW<\/em><\/strong><\/p>\n","protected":false},"excerpt":{"rendered":"