Don't Base Your Bets on Averages Alone

As an example, consider bets involving the nine at craps. Of the 36 combinations the dice can form, only 10 matter: the four adding up to nine (3-6, 4-5, 5-4, and 6-3) and the six totaling seven (1-6, 2-5, 3-4, 4-3, 5-2, and 6-1). On the average, Place bets on the nine should therefore win four times in 10. Lay bets against the nine should win an average of six times in 10. Were hit rate the whole story, Lays would be preferable.

Average hit rate isn't the sole, or primary, criterion, though. For one thing, returns differ among bets. On the nine, placing pays 7-to-5 while laying gets 2-to-3 minus an up-front vigorish. For comparison, winning with $30 placed on this number nets $42, while dropping the dealer $31 for the Lay nets only $19. Many betters balk at wagers that can lose more than they bring home.

A further distinction between Place and Lay bets outside the orb of arithmetic is that the former can be made at the table minimum, often $5 or $10, while the least casinos usually book on the latter is $30 with $1 vigorish. Some solid citizens find $31 too much to risk on a single result. Emotional influences hold as well. Many casino craps buffs believe that betting a shooter will miss-out is a craven betrayal of a living, breathing human being.

House advantage or edge, which fuses probability and payoff into a single figure, is another type of average. Edge is 4 percent for Place bets on the nine. This means that players pay the casino an average of $0.20 for every $5 wagered. A problem is that the $0.20 is invisible. Players either lose $5 or win $7. Few realize that, were the bet "fair," they'd win $7.50, so the house is raking $0.50 from the payout for every $5 bet.

Mathematically, averages fall short as adequate descriptors by ignoring the distribution of results. For instance, say you have a chance to risk $10 at odds of 1-to-1 of winning even money or 999-to-1 of winning $999. In theory, with the first option, of 1,000 hopefuls, 500 should win $10 and 500 lose $10. The second would yield one $9,990 winner and 999 $10 losers. The average is zero in both cases. But the situations differ entirely.

Parameters describing probability distributions can help players transcend mere averages in anticipating session characteristics. One such figure is "standard deviation," a measure commonly used to quantify volatility. Envision it as the average bankroll fluctuation per round of a game. To illustrate, contrast $5 bets on a column and on a spot at double-zero roulette. Edge is 5.26 percent either way. Yet the bets are hardly equivalent. The distinction jumps right out of the standard deviations, which are $6.97 for the $5 column and $28.81 for the $5 spot. Obviously, fortunes should be expected to swing more wildly when players make straight-up than column bets.

Skewness is a distribution parameter whose importance in gambling typically languishes unappreciated. Positive skew favors frequent small losses offset by occasional large wins. Negative skew, conversely, suggests repeated small wins and intermittent large losses. In either, the greater the numerical value of skewness, the further the departure from a balanced condition. For the roulette example, the skewness of the $5 column bet is 3.96 while that of the $5 spot is 29.59. The column is accordingly the choice for confidence in a modest profit, the spot for a longshot at a big payday. With the Place versus Lay trade-off, skewness is +0.4 versus -0.4 per dollar, respectively. The "plus" signifies a tendency to go for broke, the "minus" a reasonable day's pay.

Using multiple parameters to classify bets still can't project specific wins or losses in any given game. But it can suggest how various alternatives will influence the attributes of a session. A recognition thus recited by the beloved bard, Sumner A Ingmark: