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Sports Betting Articles
THE MATHEMATICS AND SCIENCE OF GAMBLING
By: jef
WHAT IS GAMBLING?
Gambling consists in risking something one possesses in the hope of obtaining something better. No one can avoid gambling, because life itself forces us to make bets on Dame Fortune. In business, education, marriage, investment, insurance, travel, in all the affairs of life we must make decisions which are gambles because risk is involved.
Many people, for pleasure or gain, also risk money on games of chance, games of skill and games which combine both chance and skill.
Games of chance are those in which there is no element of skill.
Gamblers call these "mechanical games." There are hundreds of such games , and they are the most popular form of gambling. They include lotteries, raffles, policy numbers, Bingo, wheels of fortune, slot machines, most dice games (Craps, Chuck-a-Luck, Hazard, Under and Over Seven, Beat the Shaker, etc.), and some card games (Faro, Bac¬carat Las Vegas style, etc.).
Games of skill are those in which the element of chance is com¬pletely or nearly nonexistent, such as Scarney, Teeko, Checkers, Chess, bowling, horseshoe pitching, tennis and golf.
games of chance and skill combine both elements and include most games played with cards: Poker, Gin Rummy, Bridge, Black Jack, Pinochle and many others. Sports contests such as horse racing, baseball, football, basketball and prizefights are usually thought of as contests of skill. But we must include them in this category because, from a gambling viewpoint, they all involve a certain amount of chance which sports fans know as "the breaks of the game." In baseball, for
example, a bad hop of the ball may lose the game for either team. Also, present-day bookies'' methods of handicapping or laying the odds on national sports contests are such that the element of skill plays little part in helping a bettor pick a winner.
GAMBLERS AND SCIENTISTS
Dice are the oldest of all gambling devices. Man''s earliest written records not only mention dice and dice games but crooked dice as well. Dice of one sort or another have been found in the tombs of ancient Egypt and the Orient, and in the prehistoric graves of both North and South America.
The earliest gamblers thought that the fall of the dice was controlled by the gods, and although a few of them tried to outwit divinity by loading the cubes, most of them probably considered that any prying into the matter was sacrilegious.
In the sixteenth century at least one gambler began to wonder if the scientists who were beginning to make valid predictions about other matters might not also be able to foretell how the dice would fall. An Italian nobleman asked Galileo why the combination 10 showed up more often than 9 when three dice were thrown. The great astronomer became interested in dice problems and wrote a short treatise which set forth some of the first probability theorems. His reply to the gambler was that 6 X 6 X 6 for a total of 216 combinations can be made with three dice, of which twenty-seven form the number 10 and twenty-five the number 9.
In France, in 1654, the philosopher, mathematician and physicist Blaise Pascal was asked a similar dice question by one of the first gambler-hustlers on record. The Chevalier de Mere had been winning consistently by betting even money that a six would come up at least once in four rolls with a single die. He reasoned from this that he would also have an advantage when he bet even money that a double six would come up at least once in 24 rolls with two dice. But he had been losing money on this proposition, and he wanted to know why.
Pascal worked on the problem and found that the Chevalier had the best of it by 3.549% with his one-die proposition. Throwing a double six with two dice, however, would theoretically require 24.6+ rolls to make it an even-money proposition. In practice it can''t ever be an even money bet, because you can''t roll a pair of dice a fractional number of times: it has to be either 24 or 25 rolls. Here is a calculation I have never seen in print before: The exact chances of rolling two sixes in 24 rolls are: 11,033,126,465,283,976,852,912,127,963,392,284,191 successes in 22,452,257,707,354,557,240,087,211,123,792,674,816 rolls.
This means that dice hustler De Mere had been taking a beating of
27 +% on the bet. If he had bet that two sixes would come up at least once in 25 rolls he would have enjoyed a favorable edge of .85%.
Pascal corresponded with mathematician Pierre Fermat about this and similar gambling problems and these two men formulated much of the basic mathematics on the theory of probability.
History doesn''t state how many francs Chevalier de Mere lost on his double-six betting proposition before Pascal explained why he was getting the worst of it, but I do know that nearly 300 years later, in 1952, a New York City gambler known as "Fat the Butch" lost $49,000 by betting that he could throw a double-six in 21 rolls.
Fat the Butch, although a smart gambling-house operator who has made millions booking dice games, went wrong on the bet because he figured it this way: There are 36 possible combinations with two dice, and a double-six can be made only one way-so there should be an even chance to throw a double-six in 18 rolls. Consequently, when "The Brain," a well-known big time gambler, offered to bet $1,000 that a double-six would not turn up in 21 rolls, Fat the Butch thought he had the best of it and jumped at the opportunity.
After twelve hours of dice rolling, Fat the Butch found himself a $49,000 loser, and he quit because he finally realized something must be wrong with his logic. He was, later, part owner of the Casino de Capri in Havana, and when I told him it would need 24.6 rolls to make the double-six bet an even-up proposition, and that he had taken 20.45% the worst of it on every one of those bets, he shrugged his mas¬sive shoulders and said, "Scarne, in gambling you got to pay to learn, but $49,000 was a lot of dough to pay just to learn that." "That is for sure," I agreed.
Although most of the odds and percentage problems you will en¬counter in this book can be calculated simply, a few, like the double six problem above, are more complex. Here is the formula for figuring problems of this type. To find out when the chances are approximately equal in any single event, multiply the "odds to one" by .693, the co-log of the hyperbolic log of 2. This will give the approximate number of chances, trials, rolls, guesses, etc., needed to make any event an even or fifty-fifty proposition.
For example: the odds are 35 to 1 against throwing double sixes with two dice in one roll. Multiply 35 X .693 and you find that a double-six can be expected to appear in the long run once in approximately 24.255 rolls. (The figure of 24.6+ given earlier is more exact.) To calcu¬late the approximate number required for a double event such as throwing double-sixes twice, multiply the "odds to one" by 1.678. For a triple event multiply by 2.675; a quad event, by 3.672; and a quint event, by 4.670.
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