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Best of Alan Krigman
Can you evaluate the luck factor in gambling?31 May 2010
Most casino buffs know that the house has an advantage, an edge, in its games. And that the edge is how the joints earn their dough. But few solid citizens (and not all that many bigwigs) realize that edge is a long-term affair. It's the net of casino wins minus losses over huge numbers of decisions, averaged across all the money bet. As such, it says little about what may happen on any particular coup, session, or casino visit.
More significant in the here-and-now is the volatility of the action. Volatility is a statistical concept describing the representative jumps in players' fortunes on individual events.
Picture $1 on a single spot and $1 on a 12-number "dozen" at double-zero roulette. Both have 5.26 percent edge. Technically, this means that, on the average, bosses win and bettors lose 5.26 cents per dollar at risk. But dealers don't actually collect 5.26 cents a spin from everybody who tosses a buck onto the layout.
With $1 on a single spot, on the average, 37 out of 38 folks lose $1 while the other one out of 38 wins $35. With $1 on a dozen, expectation is for 25 out of 37 players to lose $1, and 12 out of 37 to win $2. The 5.26 cents is more or less imperceptible.
These bets obviously differ significantly despite the equal edge. The distinctions are of the type that let gamblers tune their strategies to their personal preferences for risk and reward. But how? By the seats of their pants? Surely, there's a better way.
The 19th century scientist, Lord Kelvin, said "when you can measure what you are speaking about, and express it in numbers, you know something about it; but when you cannot measure it, when you cannot express it in numbers, your knowledge of it is of a meager and unsatisfactory kind."
Happily, the volatility of all casino bets can be expressed in numbers. In the simplest cases, a pair of numbers, the payoff and the probability of winning do the job. But conditions aren't always so cut-and-dried. The machines and many table games have multiple possible payoffs for a single bet, each having its own chance of success. So armfuls of numbers would become necessary. Another approach gets back to two numbers for the general case.
More universally, one of the numerical components of volatility is the "standard deviation." This is a dollar value that can be envisioned as the representative amount by which bankrolls change during the event at issue – for instance, a round or a session. For the same $1 up for grabs, higher probabilities or payoffs result in higher standard deviations and, accordingly, greater bankroll jumps; conversely for less likely or smaller returns.
The second component of volatility is "skewness." This factor quantifies the directional bias of the bankroll shifts. Positive skewness signals frequent small drops and occasional larger rises. The opposite holds for negative skewness. The magnitude, positive or negative, gauges the strength of the tendency.
The accompanying table gives standard deviation and skewness for a hypothetical jacks-or-better video poker game with an 800-to-1 jackpot, as well as several ways of betting $1 total at double-zero roulette. Edge is 5.38 percent for the video poker and a close 5.26 percent for all of the roulette options.
proposition payoff prob win std dev skewness video poker(94.62% return) various various $4.27 +165.04 one spot $35.00 1/37 $5.76 +5.86 one 3-number row $11.00 3/37 $3.24 +3.09 one 12-number dozen $2.00 12/37 $1.39 +0.78 20 spots $0.80 20/37 $0.90 -0.10 30 spots $0.20 30/37 $0.49 -1.40
The table shows the ranges of standard deviation and skewness commonly encountered. And it suggests how the values relate to the likelihood of successes and failures being of comparatively modest scope, up through those in which a handful of hopefuls strike the mother lode but most lose their shirts. These differences all occur at essentially the same edge. As the statistical songster, Sumner A Ingmark, solemnly scribbled:
'Cause the devil's in the fluctuation.
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