MIT Libraries logo MIT Libraries

MIT Libraries logo

Year 124
1984: The Mathematics of Gambling by Edward O. Thorp

Posted on May 10, 2011 in Uncategorized

Published: Secaucus, N.J., 1984

In this, his second gambling manual, Edward Thorp offers his readers plenty of practical advice, but he also spills some ink reminiscing about his days as a graduate student at UCLA. In our mutual poverty, he writes, musing on a study break with some fellow students, the conversation readily turned to fantasies of easy money. We began to speculate on whether there was a way to beat the roulette wheel.

In 1962, Thorp, then a mathematics professor at New Mexico State University, published Beat the Dealer, now considered the first card counting manual. It addressed betting techniques for blackjack only, but nevertheless made the New York Times best seller list. Much wider in scope, The Mathematics of Gambling covers everything from baccarat to backgammon.

Even before he published his first book, however, Thorp spent a couple of years as a professor here at MIT which itself would soon become a household word among professional gamblers and casino managers. Using card counting techniques pioneered by Thorp, an organized group of students from MIT played casinos across the country for a period of twenty years. The casinos eventually hired private investigators to root out these young card counters, but not before they had legally beaten the casinos to the tune of millions of dollars.

The story of the MIT Blackjack Team however fictionalized it may have been was made popular in the book Bringing Down the House, which was later adapted to the silver screen under the title 21.

The Mathematics of Gambling

Edward Oakley Ed Thorp (born 14 August 1932) is an American mathematics professor, author, hedge fund manager, and blackjack player best known as the father of the wearable computer after inventing the world's first wearable computer in 1961. He was a pioneer in modern applications of probability theory, including the harnessing of very small correlations for reliable financial gain[citation n Edward Oakley "Ed" Thorp (born 14 August 1932) is an American mathematics professor, author, hedge fund manager, and blackjack player best known as the "father of the wearable computer" after inventing the world's first wearable computer in 1961. He was a pioneer in modern applications of probability theory, including the harnessing of very small correlations for reliable financial gain[citation needed].

He is the author of Beat the Dealer, the first book to mathematically prove, in 1962, that the house advantage in blackjack could be overcome by card counting. He also developed and applied effective hedge fund techniques in the financial markets, and collaborated with Claude Shannon in creating the first wearable computer.


The incredible true story of the card-counting mathematics professor who taught the world how to beat the dealer and, as the first of the great quantitative investors, ushered in a revolution on Wall Street.

The Mathematics of Gambling

An analysis of baccarat, backgammon, blackjack, gambling games, money management, roulette and the wheel of fortune.

Beat the Market

A Scientific Stock-Market System. One of the most influential books of all time on Wall Street, whose methods launched the quant revolution of modern quantitative finance.

Beat The Dealer

The seminal book that changed blackjack forever. The father of card counting details his revolutionary point system that gives blackjack players the edge they need to win.

The Kelly Capital Growth Investment Criterion

This book is the definitive treatment of "Fortune's Formula," also described as "The Kelly Criterion", used by gamblers and investors alike to determine the optimal size of a series of bets.

Elementary Probability

A brief introduction to probability theory presenting step-by-step finite, discrete and continuous probability concepts.

Should You Bet On It? The Mathematics of Gambling

On November 9th, 2008, 22-year-old professional poker player Peter Eastgate defeated 6,843 other gamblers and became the youngest player to win the Main Event at the World Series of Poker. For his achievement, Eastgate earned $9,152,416 in cash and a spot on the list of the highest earning poker players.

Eastgate did not reach his number one spot simply through chance and speculation, however. On the contrary, casino games involve probabilities and statistics that skilled players use to guide their gambling decisions.

Three basic principles underlie casino games: definite prob abilities, expected value, and volatility index. Understanding these concepts elucidates how these games work and how people like Eastgate beat their competition.

All events in gambling games have absolute probabilities that depend on sample spaces, or the total number of possible outcomes. For example, if you toss a six-sided die, the sample space is six, with the probability of landing on any particular side one in six. Games with huge sample spaces, like poker, have events with small probabilities. For instance, in five card poker, the probability of drawing four of a kind is 0.000240, while the chance of drawing a royal flush, the rarest hand, is a mere 0.00000154.

Skilled poker players understand the sample spaces of the game and prob abilities associated with each hand. Thus, estimating the odds of a particular hand will guide their gambling choices.

Adept players are interested not only in probabilities, but also in how much money they can theoretically win from a game or event. The average amount you can expect to win is aptly called the expected value (EV), and it is mathematically defined as the sum of all possible probabilities multiplied by their associated gains or losses.

For example, if a dealer flips a coin and pays a gambler $1.00 for every time the gambler flips heads, but takes away $1.00 when the gambler flips tails, the expected value would be zero since the probability of a heads occurring is equivalent to that of a tails occurring (EV = 0.5*$1.00 + 0.5*(-$1.00) = 0). This is considered a fair game because the players have no monetary advantage or disadvantage if they play the game many times.

However, if the dealer gives $1.50 for every time the gambler flips heads, then the EV would be $0.25 (EV = 0.5*$1.50 +0.5*- $1.00 = $0.25). If this game were played 100 times, the gambler would expect to walk away with $25.

The concept of EV is important in gambling because it tells players how much money they could expect to earn or lose over all. Interestingly, all casino games have negative EVs in the long run. More commonly known as house advantage, negative EVs explain how casinos profit from gamblers.

Why, then, do professional gamblers, cognizant of house advan tage, continue to gamble if the casino is mathematically engineered to win? Additionally, how are players still able to make tens of thousands of dollars in a single game?

Though luck may be the answer for some, the mathematical answer resides in the nuanced difference between expected and actual values. EVs dictate how much a player should expect to gain in the long run, an arbitrary length of time that most gamblers do not play for. Instead, players are more interested in the actual values of each hand and the fluctuation from its EV.

The volatility index, a technical term for standard deviation, tells a player the chance of earning more or less than the EV. Using the earlier coin example, after 100 games, the player has a 68% chance of leaving the game with between -$10 and $10 and a 95% of leaving with between -$20 and $20.

Volatility index thus quantifies luck by telling players their odds of earning more than the expected value for a specific number of rounds played. High volatility games or hands have a larger variation between the expected and actual out comes and therefore, a greater possibility of winning above the EV. This possibil ity of earning above the EV is ultimately what attracts gamblers to games.

Generally, skilled gamblers assess the risk of each round based on the mathematical properties of probability, odds of winning, expected value, volatility index, length of play, and size of bet. These factors paint a numerical picture of risk and tell the player whether a bet is worth pursuing.

Still, gambling involves far more than simple mathematical properties. Gamblers use a great deal of social psychology to read their fellow players. The ability to decipher bodily cues, for instance, helps discern fellow players mental states and possibly gives a clue to the statistics of their hands.

Gambling is an art and a science; only the best players can synthesize the two to reap millions.

  • What is texas holdem poker

    What is texas holdem poker Texas Holdem Poker Texas Holdem has now become the most popular form of poker both online and in casinos throughout most of...

  • Poker online set up hands

    How To Set Up Private Online Poker Games With Friends Playing poker has never been more accessible than over the last couple of years. If the last year...

  • Single deck blackjack betting strategy

    Single-Deck Blackjack Strategy A comprehensive guide to single-deck blackjack strategy, tips, charts, and optimum plays. Learn how to win at single-deck...

  • Texas holdem online play against others

    Play Texas Hold em Online Texas holdem online play against others Texas Hold'em is a betting game where players try to get the best five-card-hand from...

  • Tropicana casino texas hold em bonus

    Casino Table Games From the classic table games of Blackjack, Craps, Roulette, and Mini Baccarat, to our poker style games such as Caribbean Stud Poker,...