How to Weigh the Risks of Alternate Casino Wagers

Gambling is about taking calculated risks in search of rewards. Although the contributing elements can be identified and evaluated,
there's no direct way to calculate what the risks are. Solid citizens find this frustrating when making bets in a casino. Civilians face the same dilemma when making decisions in industry, commerce, academia, politics, and everyday life.

The most widely-cited risk factor in casino gambling involves the player's expected net result. This is anticipated by "house advantage" or "edge," a percentage expressing what a casino reckons it earns per bet because of the disparity between payoff and odds of winning. Multiply the aggregate amount bet times house advantage to get the player's theoretical loss during a session. As an example, edge on the pass line with no odds at craps is 1.4 percent; after 20 of these bets at $10 each, a player's expected loss is $10 x 20 x 0.014 or $2.80. In comparison, the house advantage for bets on tie at baccarat is 14 percent; after 20 such wagers at $10 each, the expected loss is $10 x 20 x 0.14 or $28.

Risk also involves bankroll swings. This factor can be predicted using "fluctuation" - statisticians call it "standard deviation" or use a related term, "variance." Per-bet fluctuation symbolizes an average win or loss on every wager. Multiply this by the square root of the number of plays to obtain a value of bankroll fluctuation indicative of the swings in a typical session. To illustrate, per-bet fluctuation of blackjack with common casino rules is 1.1 units - $11 for $10 wagers; for 100-hand sessions, about an hour at a table, characteristic bankroll fluctuation would be 1.1 x 10 or 11 units - $110 for the $10 bettor. Fluctuation increases as bets become longshots. It's 5.8 units per bet on individual numbers at roulette, 29 units in a session with 25 spins. And the figures go through the roof on the slots.

A third constituent of risk involves the likelihood of short-term upswings or downswings. This is indicated by "skewness," a statistical measure that accounts both for the odds a bet will win or lose and for the payoff. Small skew means probabilities and amounts of wins and losses tend to be uniformly balanced; flipping coins for even money has zero skewness. As skew rises above zero, players lose small amounts on most bets but hope for a few moderate and large wins to snag a profit; slots work this way, exhibiting high positive skewness. Negative skew means players win small amounts on most bets and hope to avoid the damaging effects of a few big losses; betting "32 across" at craps shows this effect with skewness of ?1.49. "Buying" a 10 at craps for $21 and "laying" against the 10 for $41 highlight the contrast between positive and negative skew. For every three bets, the buy is expected to yield one $39 win and two $21 losses while the lay is expected to yield two $19 wins and one $41 loss. Expected values and fluctuations of the two bets are equal; however, skewness is +0.707 for the buy and ?0.707 for the lay.

Edge, fluctuation, and skewness can each be measured and represented numerically. In a Utopian game, edge would be low - maybe even zero or in favor of the player, fluctuation would be small enough to avoid depleting moderate bankrolls during normal downswings but large enough to satisfy all but the greediest goals during ordinary upswings, and skewness would be so negative that players would often experience sessions with few or no losses.

Singly, each of these ideals is possible. But, taken together, the better things get in one direction, the worse they get in another. Further, nobody's determined how to combine edge, fluctuation, and skewness into a single figure of merit that can be optimized mathematically to find strategies with the least risk for various degrees of reward.

Sumner A Ingmark, whose heart-rending rhymes risk rejoinder by responsive readers, recorded the riddle ruefully:

An optimum bet would remove the restriction,
That risk and reward stand in harsh contradiction,
'Tsa shame such a wager is not fact but fiction.