Should You Bet On It? The Mathematics of Gambling

Chidi Akusobi | chidi.akusobi@yale.edu February 25, 2010

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 prob­abilities, 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 prob­abilities 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 over­all. 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 advan­tage, 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 out­comes and therefore, a greater possibility of winning above the EV. This possibil­ity 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.

Further Reading

  • Hannum, Robert C. (2005). Practical Casino Math. Reno, NV: Trace Publication.
  • Packel, Edward W. (2006). The mathematics of games and gambling. Washington, DC: Mathematical Association of America.
  • Thorp, Edward (1985). The Mathematics of Gambling. New York, NY: Gambling Times.