How Important Is Edge Relative to Luck?

It's no secret. Casinos have an advantage or edge on every bet. This is what keeps them in business. But many solid citizens win anyway, raising the issue of how edge is balanced against luck.

In general, edge falls in importance the smaller it is, the fewer the rounds played, and the longer the odds -- the less likelihood of success and the higher the rewards. You may win or lose. But as edge becomes less significant, luck becomes more of a factor.

Statistics yields specific answers to questions about edge and luck. Start by recognizing that with no edge, you'd expect to break even so any win or loss could be ascribed to chance. When edge isn't zero, its significance can be gauged by estimating how long you must play before you can be confident that if you're losing, this and not bad luck is the reason.

I'll illustrate the influence of the various factors with some specifics for several wagers at single- and double-zero roulette. For the bets I'll mention, edge is 2.70 percent in the single-zero game and 5.26 percent in the double-zero version. Picture these percentages as dollars the house theoretically earns for every $100 players place on the layout.
Say you're making fix-sized wagers on three-number rows. These bets pay 11-to-1. In the double-zero game, statistical analysis shows you'd have to play 1,725 rounds to be 75 percent sure that if you're losing, edge and not pure chance is to blame. At a single-zero table, you'd have to play 6,690 rounds to reach this conclusion. Using number of rounds as the criterion, halving the edge lowers its significance relative to luck by a factor of four, other considerations being equivalent.

The extremeness of odds and payoffs has an effect because it influences the swings bankrolls typically undergo for comparable amounts at risk. As odds lengthen, keeping everything else the same, bankrolls fluctuate more wildly. The volatility tends to mask the averages, decreasing the significance of the edge.

To see how this works out, assume you're at a double-zero wheel. Betting uniformly on 12-number columns, with payouts of 2-to-1, you'd only have to play 320 rounds to be 75 percent sure you could attribute a loss to edge rather than luck. With steady bets on four-number corners, odds are steeper but wins pay 8-to-1. Now, you'd need 1,250 rounds to be 75 percent confident that a loss would be a result of edge as opposed to the mirror you broke that morning. At the limit of flat bets on a single number, hits are less frequent but they pay 35-to-1. You'd have to play 5,470 rounds to be 75 percent certain that if you were in the hole, the edge rather than the stars would be at fault.
Betting progressions are another factor, and go beyond the probabilities inherent in the games. Players often progress their bets. For instance, some increase their wagers after they win, reaching for a big score; others press after they lose, trying for a quick recovery. Progressions don't change house advantage. But they affect its significance similarly to longer odds and payoffs. That is, both raise the volatility of the game relative to the gradual bankroll erosion expected purely based on edge.

As an example, assume that instead of betting $10 per round on three-number rows at double-zero roulette, 60 percent of your wagers were $5, 20 percent were $10, and 20 percent were $25. Your average bet would remain $10, and the casino would still expect to earn just over $0.52 per round. But, these proportions of bets would let you go 2,760 rather than 1,724 rounds before being 75 percent sure that a loss was due to edge and not luck.

While casinos and their patrons play the same games, they expect to profit from them differently. The casinos depend on booking enough bets so edge is more significant than luck. Players try to operate in the opposite region -- not knowing, of course, the direction in which luck will lead them. As the poet of providence, the immortal Sumner A Ingmark, plaintively pined: