Betting Systems: Verity or Voodoo?

A primary gambling theorem holds that betting systems can't alter expectation - expected percentage gain or loss - in independent-trial games of chance. For instance, raising or lowering bets in some pattern from hand to hand at baccarat doesn't change house advantage. Likewise, big bucks on pass with full odds at craps has the same house edge as an equal amount on pass and one or more come bets with comparable odds.

Yet, gamblers galore believe strongly that how they bet affects their chances to win. And a surfeit of "systems" can be found claiming to exploit secrets about simultaneous and sequential wagering schemes the casino bosses don't want anyone to know.

Who's right: the cerebral statisticians or the superstitious solid citizens? Surprise! Both are correct. The apparent conflict only arises because they're talking about different things.

Expectation, which betting systems don't change, is an objective criterion. It's what the casino earns by virtue of its edge. And, it's a long-term average which stabilizes after tens or hundreds of thousands of decisions - many more than a single bettor would encounter even after protracted play.

Individual success, which may be influenced by systems, is a subjective goal. It may involve a host of non-monetary factors such as obtaining free meals or maximizing playing time, along with targets ranging from breaking even or showing a small profit to doubling a bankroll or hitting a lifestyle-changing jackpot. Moreover, it's a short-term effect which may be dominated by statistical properties of the game other than expectation.

To see the implications, picture seven players, all of whom gamble two weeks in a row with budgets first of $240 then of $480. Their goals are to double their bankrolls at roulette each week or lose the money trying. They'll all bet $12 a pop, but will use different betting systems. These are summarized below, with associated probabilities of winning and payoffs.
Player System
Al $12 on a single number (2.63 percent, pays $420)
Betty $6 on each of two numbers (5.26 percent pays $204)
Carl $4 on each of three numbers (7.89 percent, pays $132)
Dee $3 on each of four numbers (10.52 percent, pays $96)
Ed $2 on each of six numbers (15.79 percent, pays $60)
Fran $1 on each of 12 numbers (31.58 percent, pays $24)
Gail $6 on 1-12 and $6 on 13-24 (63.16 percent, pays $6)

The players all have the same expectation - a theoretical loss of 5.26 percent or $0.6312 per round on their $12 bets. The casino, which takes the long view, would rate them equally. Their systems are irrelevant in this respect.

Chances the players will double their money each week before tapping out differ, however, owing to single-round risk and reward characteristics and the bet-to-bankroll sizes. The probabilities of success are shown below.

Player	$240	$480
Al	48%	47%
Betty	47%	44%
Carl	45%	40%
Dee	43%	37%
Ed	39%	29%
Fran	33%	20%
Gail	11%	2%

These probabilities suggest that to double a stake before losing it, chances of success improve as bets become a) greater longshots with larger payoffs but steeper odds, and b) higher fractions of the starting bankroll. Players having other criteria - say, extending a streak of winning games without regard to amount, minimizing the chance of going belly-up, or testing the air in a high-limit pit - might find this strategy disastrous.

Betting systems can be optimized for any specified gambling goals. High likelihood is no guarantee, of course. And, don't forget the insidious law of unintended consequences. Raising the chance of meeting a specified set of goals may have an unpleasant downside, like excessive loss when things go wrong, too long a required playing time, sacrifice of a desirable fall-back position, or sneers from dealers you're trying to impress.

It's as Sumner A Ingmark, poet of intended nonconsequences, said:
State your goals precisely, clearly,
What's unsaid may cost you dearly.