How much money should you bring to a casino, or equivalently, how much should you bet per coup, given such and such a bankroll? Affordability, of course, is a major factor. Folks shouldn't gamble thinking from the outset they'll go belly-up. But there's a chance this'll happen, and risking the rent or mortgage money isn't exactly a sound fiscal or recreational policy. After affordability comes the question of gambling objectives. Some solid citizens picture this in terms of a "win goal." These individuals ostensibly won't be happy unless they earn the profit they have in mind. So they persist until they either hit this target (at which point they often figure they can do better so they revise the goal upward as opposed to quitting), bust out, or get dragged away from the game to eat, sleep, or hop on the bus.
The other, generally more common and realistic, gambling target involves time. Nobody wants to exhaust a stake in short order then spend the next few hours watching the other suckers lose their money, too. People instead tend to envision getting two to four hours' action during a visit ?? figuring this gives them a dose of excitement along with a shot not only at making a good buck but also getting a nice comp for the all-you-can-eat buffet.
Both of these classes of gambling goals have been studied extensively. Not by monitoring players and forming conclusions based on the observations. Rather, by performing statistical analyses and solving the equations. The pronouncements made this way prove to be reasonably reliable, despite the idiosyncrasies of particular games and bettors. So, when the more credible gurus say you have a 90 percent chance of surviving for three hours or longer in a certain game, betting $5 on a $500 bankroll, you can believe it and use this kind of information to guide your play.
As an example, consider two hypothetical games with the chances of losing and winning various amounts shown in the accompanying table. These games have the same 4.5 percent house edge. They also have equal volatilities or average bankroll swing characteristics (the math mavens would say that the "standard deviations" in these games are 1.6 times the bet in either case).
Hypothetical games with equal edge and volatility
but different skewness, payoffs for bets of $1
Despite equal edges and volatilities, the games clearly differ. The low-skew version loses more often and has a moderate chance of a smaller top payoff. The high-skew game has more intermediate returns and a lower probability of a bigger top win. These properties are measured mathematically by the "skewness" index. It's 1.7 in the first instance and 6.1 in the second. The math used to estimate the chance of surviving sessions of various durations accounts for edge, volatility, bet size, and bankroll but ignores skew. This simplification does not invalidate the predictions under most practical conditions.
Say these games were slot machines. You want to play, betting $1 per round on a bankroll of $100. Assume you average about 600 spins per hour. Analysis gives your chance of being in the game for four or more hours as only 33 percent. Computer simulations of 100,000 sessions for each machine confirms the prediction, showing that 32 percent of all players were still active after 2,400 spins. You'd get a better shot at this much gambling time by betting less or starting with more money. A $100 bankroll with bets of $0.50 or a $200 stake with $1.00 wagers is predicted to give an 82 percent chance of four hours' action. The $100 bank and $1 bets might be more appropriate if you only wanted two hours' gambling time, affording 70 percent chance of survival.
Sure, luck determines what happens in any particular instance. But the amount of luck you need to reach your goals depends on the extent to which you have to buck the laws of probability to get there. Here's how the poet, Sumner A Ingmark, put it:
Those who recognize and heed,
The laws of chance will best succeed.