Even Fighting Edge, You Can Get Close to the Ideal... At a Price

11 September 2002

By Alan Krigman

The "rule of inverses" puts an easily estimated ceiling on the chance of increasing a bankroll by any given factor, versus going bust. Simply stated, the probability of success never exceeds the inverse of the multiple by which you wish to increase your stake.

Say you start with $500 and won't quit until you multiply it by two (win $500 and finish with $1,000), or bite the dust. The likelihood you'll succeed may get close to but not over one out of two, which is 1/2 or 50 percent. What about putting up $200 to win $800? The probability of getting from $200 to $1,000 can't get above one out of five, which is 1/5 or 20 percent. In a more conservative vein, suppose you're content risking $1,000 to earn $100. That's multiplying $1,000 by 11/10 to quit with a total of $1,100. Your chance approaches a strong 10/11 or 90.9 percent.

The limit set by the rule of inverses is the ideal. It holds when the house has no edge. In such a gamble, the nitty-gritty would be irrelevant. Bet size wouldn't matter. Nor would volatility (think of volatility as the normal bankroll fluctuation per round - math mavens call it "standard deviation"). Unfortunately, edge does rears its ugly head, not only becoming an issue on its own, but also bringing bet size and volatility into the equation.

Every situation is unique, so no single handy-dandy parameter predicts how different real games diverge from the ideal. But you can get an idea about how your options affect your outlook, and adjust your play to suit your personal preferences.

Make believe you're playing blackjack, starting with $500 and itching to win $1,000 and walk with $1,500. The rule of inverses limits the chance you'll triple your money from $500 to $1,500 before hitting the canvas, to one out of three or 33.33 percent.

Envision four scenarios. 1) Basic Strategy with equal bets in every round. 2) Typical departures from Basic Strategy with equal bets in every round. 3) Basic Strategy with a betting progression involving one unit half of the time, two units a fourth of the time, and four units the other fourth. 4) Typical departures from Basic Strategy with the 1-2-4 three-stage betting progression.

House edge following Basic Strategy in a moderately liberal game is about 0.5 percent. Typical departures from Basic Strategy might raise this to 1 percent. Standard deviation in blackjack is 1.13 times the bet; this characteristic round-by-round fluctuation accounts for a preponderance of one-unit wins or losses, interspersed with occasional 1.5-unit wins on naturals and two-or-more-unit exchanges on splits and doubles. The 1-2-4 progression yields an average wager twice the base unit and a standard deviation equal to 2.65 times the average bet.

The accompanying chart shows the probability of tripling a $500 bankroll before tapping out for the four scenarios, with bet levels - flat or average - of $10, $50, and $100. You can see the influence of the various factors. The chance of reaching a profit goal improves as edge decreases, volatility rises (here because of progressive wagering, in other cases due to jackpots and other high-payoff propositions), and bet size increases.

Unfortunately, gambling has a way of taking as it gives. Pumping up bet size and volatility improves the probability of fulfilling your fantasy before floundering. However, if you don't hit your target and quit, the same tactics make your initial stake less apt to last. Basic Strategy players who bet $10 flat have only 21.4 percent chance of tripling a $500 poke; but, if they miss, they have 98.7 percent chance of being in action for 300 rounds. Were the same solid citizens to start betting $50 and progress to 100 and 200, they'd have nearly the ideal 33.33 percent chance of meeting their win goals; if they fail, though, their prospects of surviving 300 rounds are only 8.4 percent. The beloved bettors' bard, Sumner A Ingmark, viewed the duality of gambling like this:

Bettors who think they can have it both ways,
Wander away from the games in a daze.