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You can get lucky chasing losses, but the more you play, the less the chance

31 January 2011

Casinos generate their earnings by nibbling off small percentages of the money players wager. Say, for instance, you're a roulette buff and always bet $5 on each of three rows – $15 total per spin. Edge, assuming a double-zero game, is 5.26 percent; the bosses therefore figure your action per spin is worth 5.26 percent of $15 – just under $0.79. During a three-hour session on a certain visit, you'd get about 120 spins, making you a $0.79 x 120 or $95 customer.

The $0.79 and $95 are "theoretical" – averages over large numbers of bets made by hosts of players, each of whom wins or loses some amount. Individuals really only care about the results of their rounds and sessions. You don't notice the $0.79 and wouldn't gamble if you always lost $95 in three hours. In practice, after this much time, your chances are 50 percent of losing more than $95 and 50 percent of losing less, including 37 percent of winning. This, because edge isn't the only factor affecting particular players' outcomes. Volatility is the other major determinant.

Edge goes one-way. From bettors to bosses. On the average, for every spin with $15 at risk on double-zero roulette, players pay the house $0.79. And those $0.79 increments mount linearly. After 25 spins, the expected total is 25 x $0.79 or $19.75; after 100 spins, the casino figures 100 x $0.79 or $79; and after 1,000,000 spins, the joint estimates 1,000,000 x $0.79 or $790,000.

Volatility is bidirectional. When nothing hits in any of the three $5 rows, your bankroll shrinks by $15. On a hit, you're paid $55 for the row in question minus $10 for the two misses, raising your fortune by $45. For one spin, the characteristic bankroll jump under these conditions (math wonks call it "standard deviation") is $25.50 up or down. For multiple spins, one standard deviation defines the upper and lower ends of a range covering 68 percent of all net bankroll changes. And it's balanced so half the remaining 32 percent is below and half above this range.

As with edge, the amount represented by volatility gets bigger as you make more bets. But because bankroll jumps may be up or down, there's some cancellation and the cumulative value is less than simply the number of spins times $25.50. Standard deviation accordingly grows more slowly than does edge. After 25 spins, it doesn't reach 25 x $25.50 or $637, but 5 x $25.50 or $128; after 100 spins, it's not up 100-fold to $2,550 but to 10-fold to $255; and after 1,000,000 spins, the value isn't 1,000,000 x $25.50 or $25,500,000 but 1,000 x $25.50 or $25,500.

The higher rate at which edge increases relative to volatility explains why casinos depend on long-term averages while solid citizens pin their hopes on the short run. For one spin, the effect of volatility is much greater than that of edge. Minus $15 or plus $45 swamp the $0.79 penalty. After 1,000,000 spins, edge dominates – $790,000 down as opposed to a standard deviation of $25,500 up or down. At some number of spins, the two are equal. For the $15 per spin double-zero roulette example, the crossover is 1,044 spins – 26 hours assuming 40 spins per hour. At this point, edge represents an average loss of $824.21, and one standard deviation a swing of $824.21 above or below this average. This means players would have 16 percent chance of being more than twice $824.21 in the hole and an equal 16 percent chance of having risen more than $824.21 above the average to find themselves with a profit.

In case you're wondering, you needn't play 1044 spins for edge to gain enough on volatility to negatively influence your chances of earning a profit. It's a matter of degree. The accompanying table shows the effect for sessions of 40, 80, and 120 spins betting as in the example.

Edge rises faster than volatility with extended play

spins     loss due to edge      standard deviation      probability of being even or ahead
40            -$32                 $161                         42%
80            -$63                 $228                         39%
120            $95                 $279                         37%

The casino views all of this oppositely. The bosses want the most action possible, over lots of patrons and long periods, so the effect of volatility will be insignificant compared to that of edge. After 5,622 spins, the chance of players as a group being ahead is only 1 percent. After 9,950 spins, it's a mere 0.1 percent. After a million spins, the casino has statistically expected earnings due to edge of $790,000 with 99 percent probability of being between $715,000 and $865,000, and virtually no chance of having lost to the hoi polloi..

If you get behind in a game, you may chase your losses and keep playing. Or, if you get ahead, you may think you can go on forever. It may work. You may get lucky, hit a hot streak, and strike it rich. But the laws of the universe, as applied to edge and volatility, say that the longer you play, the less your chances of success. The punter's poet, Sumner A Ingmark, put it like this:

Gamblers who cannot decide when to finish,
Find that their chances of profit diminish

Alan Krigman

Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.
Alan Krigman
Alan Krigman was a weekly syndicated newspaper gaming columnist and Editor & Publisher of Winning Ways, a monthly newsletter for casino aficionados. His columns focused on gambling probability and statistics. He passed away in October, 2013.