Craps wouldn’t be a game of chance if you knew the outcome of the roll in advance. Why? By definition, the outcome of any game of chance must be uncertain. The branch of mathematics that deals with this uncertainty is called Probability.
The science of Probability was born in the seventeenth century when the Chevalier de Mere, a French nobleman who enjoyed a good gamble as much as the rest of us, began posing questions about the probable outcome of certain dice wagers to prominent mathematicians. The Chevalier had made a considerable amount of money on even money bets – wagering that the number six would be rolled at least once in four rolls of a single die. He did so well with this wager that he sought to expand upon it by wagering that if two dice were rolled the double-six would show up at least once every twenty-four rolls. Needless to say, his math failed him and he lost his money just as quickly as he won it with the single die wager.
Baffled by his misfortune, the Chevalier took his problem to the mathematician Blaise Pascal. Pascal, in turn, consulted with another mathematician, Pierre de Fermat, and the two of them solved the double-six problem and laid the foundations of the Probability branch of mathematics.
By the way, the rule of mathematics used to solve this problem was the Multiplication Rule of Independent Events. It states simply that “For independent events, the probability of all of them occurring equals the product of their individual probabilities.
Okay, let’s go back to the two-die problem. Each die has six sides. The number six appears one time on each die. So the odds of a six showing up on any one roll are 1-6. But when you add the second die in things get more complicated. Since each die operates independently of the other you must calculate the probability of all possible outcomes. You do that by multiplying the independent probabilities (1-6 and 1-6) together. In this case, the result is the correct odds of the double-six rolling: 1–36. The Chevalier’s mathematical mistake – multiplying six (the number of sides on a single die) times four (the number of rolls he planned to make) resulted in his one roll in twenty-four mistake – and cost him a fortune.
Now, let’s take the Multiplication Rule of Independent Events and apply it to some of the things we talk about on the board from time to time. Some of you have been at the table with me when I’ve tossed five, six, seven, and on one occasion – eight consecutive Horn numbers. What are the odds against that happening? I’ll walk you through the math.
There are six ways to win on the horn bet – one way each on the two and twelve, and two ways each on the ace-deuce and yo. There are thirty-six possible combinations of the dice. So the odds of a horn number rolling on any given toss are 6-36, which reduces down to a nifty 1-6. From there it’s just simple mathematics to calculate the odds of consecutive hits on the horn occurring.
The odds of two consecutive horn numbers showing: 1-36 (6X6)
The odds of three consecutive horn numbers showing: 1-216 (36X6)
The odds of four consecutive horn numbers showing: 1-1296 (216X6)
The odds of five consecutive horn numbers showing: 1-7776 (1296X6)
The odds of six consecutive horn numbers showing: 1-46,656 (7776X6)
The odds of seven consecutive horn numbers showing: 1-279,936 (46,656X6)
The odds of eight consecutive horn numbers showing: 1-1,679,616 (279,936X6)
That’s correct. Eight horn number in a row will happen just once in every 1.68 MILLION tosses of the dice. With figures that staggering, it is no wonder the math crowd says this dice setting thing doesn’t work. And it’s no wonder that the rest of us say we’ve got all the proof we need that it does.
Take, for example, a group session a dozen of us were playing down at the Golden Nugget many years ago. A gal we called “Dice Chick” set the dice in a configuration that favors Horn bettors and managed to toss thirteen consecutive Horn numbers on the come out. Trust me. When Dice Chick got out around toss number nine none of the players in attendance were worried about the odds of her tossing number ten. We were all too busy counting our money.
Understanding the odds of the games you play will help you make smarter decisions at the table. The Horn is generally a bad bet – unless you can toss a 2, 3, 11 or 12 at a higher than random rate. That, in turn, can lead to the best math of all. The math exercise that occurs when you say, “Color coming in.”