You’re headed for the casino. You have $100 in your fanny pack and a premonition you’re gonna leave with $1,000 and a bellyful of U-Krak-Em crab legs from the buffet, compliments of the house.
How plausible is the hope that you’ll achieve your monetary goal, multiplying your stake by 10 rather than losing it? Absent house advantage, the “inverse rule” gives an elementary guideline. The chance of multiplying a bankroll by some number before losing it equals the inverse of that number. To go from $100 to $1,000, you multiply by 10 so the probability is 1/10 or 10 percent.
The inverse rule gives you the ideal. The best shot you can get gambling with no house edge. But casinos that stay in business don’t offer games with this feature. Inquiring minds accordingly want to know how much the edge dims the outlook. Unfortunately, the answer isn’t quite as simple as the question implies.
Say you play double-zero roulette with $10 per spin on a single spot. Edge is 5.26 percent. The chance of reaching $1,000 drops from the abstract 10 percent to a concrete 8.64 percent. Instead, make believe you bet $10 per spin in the same game but put $2 on each of five spots. The edge stays at 5.26 percent. However, your chance of getting from $100 to $1,000 falls to 3.96 percent.
There’s more. Assume you raise your total bet from $10 to $20, sticking with the same propositions. Edge is still 5.26 percent. With $20 on a single spot your chance of hitting $1,000 is 9.3 percent, up from the 8.64 percent at $10 per spin. Putting $4 on each of five spots, your prospects are 6.49 percent; compare this with 3.96 percent for $2 on each of the spots.
The four situations, $10 or $20 on one spot or $2 or $4 on each of five, have the same edge but different volatility. This factor is a gauge of characteristic bankroll fluctuations; it accounts for what you can win or lose per round, and the associated probabilities. Mathematicians measure volatility in terms of “standard deviation.” Values of this quantity are $58 and $115 for $10 and $20, respectively, on one spot; they’re $24 and $49 for $10 and $20, respectively, spread evenly over five spots.
What about the other part of your casino goal? The coveted comp for the all-you-can-eat buffet. Casinos award prominent patrons these gratuities based on the edge and the gross amount bet. At roulette, the bosses return about 1.7 percent of the gross wager. You can figure on making 50 bets per hour. At $10 total per spin, you’d accordingly have to play for an hour and a half to get 75 rounds, a “handle” of $750, to earn a comp worth $12.75.
Say you have a $200 gaming budget and bet the $10 as $2 on each of five spots. You have 60 percent chance of being in action for at least 75 spins. Alternately, if you raise the volatility by betting the $10 on a single spot, hoping to grab a bigger win, your chance of surviving for this duration drops to 29 percent. Here, increasing volatility works against you by lowering the likelihood of a long enough session to earn the required credits.
Solid citizens can exercise a limited degree of control over the house advantage through their choices of games and playing strategies. Notwithstanding edge, however, bettors have lots of opportunity to tweak their sessions by adjusting volatility. The primary tuning mechanisms are concentrating risk and raising bet size to boost volatility, and conversely to lower it. Roulette offers a clear example of how this can be done, but the idea can be applied to most other table games and many machines as well.
High volatility raises the chance of reaching a win level with a particular bankroll but diminishes that of lasting for a long session. Low volatility has the opposite effect. For players whose stashes won’t normally be mistaken for Fort Knox, it may well boil down to a choice of copious cash or bounteous buffet. It’s up to you. And to fate, of course. In pondering such puzzles, the perceptive punter’s poet, Sumner A Ingmark, noted:
Gamblers having great facility,
Know both edge and volatility.