How Does Bankroll Fluctuation Affect Your Fortunes?

Few gamblers appreciate a large upswing when they're fortunate enough to get it. Fewer are prepared for a downswing of depth and duration quite normal for the game they're playing and the bucks they're betting. Worse yet. Proficient players often wonder why they're losing while bozos making sucker bets or using bogus "systems" are raking in the dough. And novices are sure that gambling is strictly luck, with no skill or knowledge required.

One cause of these anomalies is that different factors dominate long and short runs. "Edge" - or "house advantage" - is the best-known gambling parameter. But it most reliably predicts long-term results, for instance when casinos balance their books at the end of major accounting periods. "Bankroll fluctuation" - statisticians call it "standard deviation" - is less familiar. But it most dependably characterizes short-term performance, like that of individual gamblers during normal or even Marathon play.

Both elements contribute to gambling results. In games with high-payoff longshots, skewness also becomes important; however, for now, I'll focus on even-money bets where skewness isn't relevant. To illustrate the idea, consider a baccarat session comprising 400 hands, each with a $10 bet on "Banker."

Under most baccarat rules, the edge on Banker is 1.09 percent or 0.0109. Multiply this by the wager and total number of hands to get the bettor's theoretical session loss. For the example cited, this would be 0.0109 x $10 x 400 or almost $44.

The fluctuation on a single Banker bet is $0.93 per $1 wagered or 0.93. To get overall session fluctuation (you'll need a calculator here, but it won't hurt much), multiply this ratio by the bet and the square root of the number of hands. For the example being used, the square root of 400 is 20; the session fluctuation is therefore 0.93 x $10 x 20 or $186, far more than the edge effect.

With these two factors, you can estimate reasonable buy-in and win figures. Here's how to do it (keep that calculator handy):
1) Start with the theoretical session loss due to edge. For this example, it's $44.
2) Multiply session fluctuation by three. Less than 5 percent of all games drop further below the theoretical loss than triple the session fluctuation. Here you get $186 x 3, or $558. So $558 + $44, about $600, is enough buy-in to outride normal downswings in 400 $10 bets on Banker in baccarat.
3) Last, multiply session fluctuation by 1.41. Less than a quarter of all games will exceed the theoretical loss by more than this value. For the baccarat case the result is $186 x 1.41, or $262. Therefore, wins of $262 - $44 or about $220 shouldn't be expected more than 25 percent of the time.

The example shows that fluctuation has a greater impact than edge on individual players in ordinary sessions. The opposite holds for casinos over extended periods. Say a big operator averages 25 million $10 Banker bets in a year. Theoretical earnings from edge are 0.0109 x $10 x 25,000,000 or $2,725,000. The square root of 25,000,000 is 5,000, so fluctuation is 0.93 x $10 x 5,000 or $46,500, far less than the edge effect. From rule #2, the casino has less than 5 percent chance to be either under or over 3 x $46,500 or $139,500 of $2,725,000. It therefore has a 90 percent chance of netting between $2,585,500 and $2,864,500 on this bet.

Try the calculations for other games and playing times. You'll be in the ballpark with these starting points. Blackjack: edge is 0.5 percent with basic strategy, 1 percent with hunches; fluctuation ratio is 1.2. Flat "pass" or "don't pass" at craps: edge is 1.4 percent; fluctuation is 1.0. Red or black at roulette: edge is 2.6 percent; fluctuation is 1.0. If you use 25, 100, 400, and 900 rounds, square roots are 5, 10, 20, and 30, respectively.

Sure, you can cruise into a casino and work one wager up to a retirement annuity. But you've a much better chance of success starting with a stake and quitting having reached a profit which the laws of probability say are consistent for your game. Sumner A Ingmark, the bard of the bankroll, probably put it best:

When stories of success are aired,
They show luck finds the well-prepared