The Calculus Diaries - Jennifer Ouellette [29]
The key difference is if the shooter rolls a 12. In that case, a don’t-pass bet will neither win nor lose; it would be a “push.” This is simply a means of maintaining the house advantage: Three numbers are losers while two are winners on the come-out roll if you place a pass bet. In contrast, two numbers are losers and two are winners if you place a don’t-pass bet on the come-out roll. One might be tempted to conclude, therefore, that the odds of winning the come-out roll with a don’t-pass bet are 50/50. One would be mistaken. Probability is more complicated than that, even for a relatively simple game like craps, which is why the field has fascinated scientists and mathematicians for centuries.
CHANCE ENCOUNTERS
Among the first to analyze games of chance with an eye toward odds and winning strategies was a sixteenth-century physician, astrologer, and mathematician named Gerolamo Cardano. Born in 1501, his was not the most auspicious of beginnings. His mother, having already borne three children and clearly being fed up with parenthood, tried to abort him with a brew of wormwood, barleycorn, and tamarisk. Gerolamo survived but promptly contracted bubonic plague when he was just a few months old—usually a death sentence at the time, particularly for an infant. Astoundingly, he survived that, too. (His wet nurse and three half-brothers perished.)
His father, Fazio, wanted the teenage Gerolamo to study law, but the boy longed to study medicine instead. He initially supported his studies by tutoring others in geometry, alchemy, and astronomy, as well as casting horoscopes. (Astrology and alchemy were still considered legitimate fields of study.) But then he developed a taste for gambling and found he had a gift for beating the odds. He quickly amassed winnings of 1,000 crowns, more than enough to pay for his education, and in 1520 began writing a treatise, The Book on Games of Chance, which he kept revising right up until his death.
Cardano was a better gambler than a physician, it seems—or rather, he lacked the business acumen to market himself to prospective patients. He struggled mightily to support his family early in his career, and soon found himself resorting to gambling again to make ends meet. Eventually Fortune seemed to smile on him: He published a series of successful books and by 1550 became the renowned physician he’d always dreamed of being.
If only he hadn’t had children. Cardano’s appalling offspring were a trio of bad seeds whose behavior would make Caligula blush. His daughter Chiara seduced her older half-brother, Giovanni, at the age of sixteen, became pregnant, aborted the fetus, and continued to philander even after her marriage, eventually contracting syphilis. That same brother was later convicted of poisoning his wife; Cardano spent a fortune on his defense, to no avail. Giovanni was summarily executed, most likely deservedly so. The younger son, Aldo, became a torturer for the Spanish Inquisition, testifying against his own father so that Cardano briefly landed in jail. Cardano finally died in September 1576, penniless and quite mad, having burned more than half of his manuscripts before shuffling off this mortal coil.
Among his surviving manuscripts was The Book on Games of Chance, finally published in 1663, almost a century after Cardano’s death. By that time, others had replicated and out-paced Cardano’s analysis, but the beleaguered physician with the rotten luck deserves his minor place in the annals of probability theory. In chapter 14, titled “On Combined Points,” Cardano laid out what we now know as the law of the sample space. The sample space is simply the set of all possible outcomes of a random process (like the roll of the dice or flipping a coin). Cardano reasoned that the probability of winning a roll of the dice, for example, is equal to the proportion of winning outcomes. A die can land on any one of its six sides, and those six potential outcomes make up the sample space. Place a bet on one such number, and