Why Skew May Be More Important to Casino Gamblers than Edge

30 August 2004

By Alan Krigman

Many casino aficionados choose games based on "skewness," although they don't know what it is or even that they're doing it. Skewness puts a number on the trade-off between a good chance at a small profit and a low probability of a big score. Assuming some point on the spectrum between the extremes of these results fits your personal objectives, this statistical quantity also indicates how lucky you must be to achieve your goals.

Skewness explains why so many bettors ignore all the conventional hoo-hah about edge and aren't terribly interested in volatility, either. The former because edge doesn't have an evident impact on the results typical recreational players experience during sessions, casino visits, or lifetime gambling careers. The latter because few punters have the discipline to quit when they lose what they thought was sensible before they left home or win enough to gain bragging rights when they head back.

You can get an idea of the magnitudes involved, and how they change with probabilities and payoffs, from the accompanying data. It shows skewness for bets of $30 a spin at double-zero roulette, when the money is spread out eight different ways, from the entire amount on a single spot to $1 on each of 30 numbers.

Skewness for eight alternate ways of spreading $30 at roulette

bet	prob of winning	payoff	skewness
$30 on one spot	1/38	$1,050	+5.92
$15 on each of two spots	2/38	$510	+4.00
$10 on each of three spots	3/38	$330	+3.12
$6 on each of five spots	5/38	$186	+2.18
$5 on each of six spots	6/38	$150	+1.88
$3 on each of 10 spots	10/38	$78	+1.08
$2 on each of 15 spots	15/38	$42	+0.43
$1 on each of 30 spots	30/38	$6	-1.32

All the options have the same edge, 5.26 percent. So, were this the only or the primary criterion for selecting among bets, they'd be equivalent. Yet, you don't have to be a Nikolai Ivanovitch Lobatchevskii to realize they're inherently different.

Say you bet $30 per spin on single spots, the $1,050 payoff means you could leave with substantial earnings if you got lucky. But the 1-in-38 chance of winning implies you're apt to bust out in short order on a moderate bankroll. The other extreme, distributing the money as $1 on each of 30 spots, won't get you rich at $6 per win; however, the 30-in-38 probability of success suggests you'd be very unlucky not to survive at the table for sufficient time to satisfy your craving for action, earn a modest profit, and get a comp for the all-you-can-eat buffet. These effects, and what's in between, are reflected in the skewness.

Near zero, skew signifies wagers you anticipate winning about as often as losing, typically for close to even money. Positive skew (the plus sign) applies to bets you're apt to lose more often than you win, but payoffs equal or exceed the amounts at risk. Negative skew (the minus sign) designates bets you're likely to win more frequently than you lose, but what you put at stake exceeds what you can pick up. Either way, higher numerical values mean stronger tendencies.

Most casino patrons today prefer bets that turn out to have moderate positive skews. They typically want to avoid extreme longshots that will wipe them out too soon, but they're still interested in spinning flax into gold. Traditional slot machines and jackpot-oriented table games fall into this compromise realm using payoff schedules that combine low-probability high returns with regular small rewards. Parceling money around a roulette layout can achieve the same result, as the data indicate.

You can achieve negative skew, if this is your preference, at roulette by increasing your bet spread as indicated, and at craps by making lay or multiple place/buy/come bets. The new multi-line machines offer slot players a similar opportunity. Betting one coin on each of nine lines, for instance, may do it.

Some solid citizens go to the casino with the attitude that all bets are pretty much the same -- gambling is gambling. Others see the sagacity in this stanza by the songster, Sumner A Ingmark:

A goal and a plan to attack it,
By gut or with math that can back it,
Mark gamblers who prove they can hack it.