The Ultimate Guide to Casinos



One of the most important aspects of managing a bankroll is planning for downswings and upswings in the course of casino play. Every session is different. There will be successes and failures. Sometimes play will be flat, with neither gains nor losses. Occasionally big wins will occur and, for those who are vigilant, potential financial catastrophes should be few and avoidable.

The term used in casino gaming to describe how short-term results differ from expected outcomes is “Variance.” Statistically, Variance is related to a game’s volatility, and it can be defined by the mathematical equation,

V = σ2 ,

where “V” stands for Variance and “σ” is the game’s Standard Deviation. (Note: the formula for σ and a chart showing its values for various games appears in the section on “Volatility,” elsewhere on this site.)

Although the equation always results in a positive number, it should be noted that Variance can be used to quantify the likelihood of both positive outcomes (winning more than anticipated) and negative ones (losses exceeding expectations). In practice, it is quite useful in managing one’s money to guard against potentially ruinous sessions.

Developing a Bankroll Strategy

We know that almost all casino games favor the House, and over the long term players are expected to lose at a rate equal to the House Edge. However, knowing the Variance of a game can help quantify its short term risk, which is much more useful to a player than knowing the House Edge. That’s because a game’s volatility dominates results over the short haul.

Take for example the player who brings $1,000 to a Blackjack table with favorable rules, where the built-in advantage for the House is just 0.4%. Based solely upon House Edge, the expectation might be that $4 will be lost if the entire bankroll is played through. But when the mathematics of Variance is applied, calculating a Standard Deviation of 1.32 for this particular game, some very different scenarios and risks can be identified.

Suppose the player chooses to wager the $1,000 by making $10 flat bets each hand, never varying the amount wagered; Variance indicates that 10% of the time, playing a session of 100 hands, the real expected loss is $150. What’s more, there’s a 0.01% chance that $430 will be forfeit—a far cry from $4.

On the other hand, using the $1,000 to flat bet $2 each time for 500 wagers, Variance calculations yield a 10% chance of losing $66, and an expected $190 loss just 0.01% of the time. Clearly, smaller bets over a longer session must be seen as the less risky course of action, although the trade-off is that any projected winnings will be reduced proportionately. Beyond the numbers, it all comes down to how much a player is willing to risk.

Variance at Work

Because Variance swings in both directions, it can be the player’s best friend or worst enemy. Games with a high degree of volatility and thus a large Variance may not behave “normally” in the short term. A Video Poker machine can “get hot” or “go cold.” Craps shooters may “get on a roll” or “lose their touch.” Variance explains why Keno players will mark blocks of consecutive numbers and Roulette players will double up on a winning number, looking for it to repeat.

Because of Variance, all casino games appear to evidence a kind of “Streakiness” at times, as discussed in greater detail in another section on this site. Knowing this, players may wish to seek out high Variance games, play them for short sessions and try to take advantage of any streaks that occur. But they should also be aware of the downside of doing so, as evidenced time and again in casino history.

Perhaps the best example of Variance wreaking havoc is the so-called “Monte Carlo Fallacy.” It was made famous on August 18, 1913, when the croupiers at a Roulette table in Monaco’s Monte Carlo Casino spun 26 Black numbers in succession. Gamblers that day lost millions of francs betting on Red, erroneously reasoning that the streak had to be “balanced out” by a similarly long streak of Red numbers.

Perhaps they were right, but not in the short term. In 1943, the color Red won 32 consecutive times at a U.S. casino. Of course, most players there began wagering on Black long before the streak ended. That’s extreme Variance in action.