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# Is There a Way to Evaluate Luckiness in the Casino?

12 July 2004

Can you put a number on how lucky do you have to be to win in a casino? Bet-by-bet, the value of probability might do the job. For instance, a Buy bet on a four at craps is a three-out-of-nine proposition. For a Lay bet on the four, chances are six out of nine. Buys therefore seem to need twice the luck as Lays. But this ignores payoffs. With the vigorish, \$21 out on a Buy nets \$39 while \$41 on a Lay yields \$19. So, the greater chance risks more to gain less. Luck should somehow account for this.

The expectation associated with a bet, or its percentage forms -- house advantage or edge -- combines probability with payout and is often used to gauge the luck needed to beat the bosses. The idea is that higher negative expectation requires more luck to overcome. While true in principle, in practice this is only valid after a multitude of decisions are resolved. The kinds of action the joints chalk up over weeks or months but more than most solid citizens encounter in their lifetimes, let alone single sessions or casino visits. Further, the luck needed to win with a \$20 Buy and \$40 Lay seem different, but expectations are each to lose \$1.

Another candidate for evaluating luck, volatility, is an amalgam of probability and payout that anticipates sizes of bankroll fluctuations during a game. The limitation is that volatility is bidirectional and doesn't distinguish up from down. Small swings mitigate bad luck by keeping you solvent, but stave off good luck by cutting your prospects for earning enough to be satisfied. Conversely for big fluctuations. Further, in comparing \$20 Buys and \$40 Lays, characteristic bankroll shifts (officially known as "standard deviations") are \$28 per decision in either case.

Skewness, a third parameter incorporating probability and payoff, may hold more promise for evaluating luck in the casino. Bets which win frequently but generate small payoffs are negatively skewed (minus values). And, the easier it is to grab lower sums, the more negative. Conversely, bets which rarely hit but pay well are positively skewed (plus values). Here, magnitude increases as chances shrink and payoffs grow. The implication is that you'd have to be unlucky to lose with negative skewness, and more negative denotes worse luck. Conversely, you'd need good luck to win with positive skewness, and more positive suggests you need better luck.

Several alternate double-zero roulette bets illustrate the concept. You're favored with \$5 on each of two columns since your \$10 affords 24 ways out of 38 to pick up \$5; skewness is -0.54. You're favored more to earn less if you bet \$1 on each of 10 three-number rows, having have 30 ways out of 38 to receive \$2; skewness is -1.42. Both offer good chances for small returns so skew is negative; because the rows bet is easier to win but pays less, skew is numerically larger. Conversely, you're bucking the odds betting \$10 on one row, with three ways out of 38 to win \$110; skewness is +3.12. Even more difficult, \$10 on a single spot, has one way out of 38 to collect \$350; skewness is +5.92. Since both are longshots with large returns, skew is positive; the spot bet win less often but pays more, so skew is numerically larger.

Skewness likewise distinguishes the \$40 Lay and \$20 Buy at craps. The former has skewness of -0.71, the latter +0.71. Moderate bad luck to lose the Lay, comparable good luck to win the Buy.

Relatively neutral situations, such as even-money bets with close to 50 percent chance of winning, have almost zero skew. For example, flat line bets at craps, with 49.29 percent chance of winning even money, have skewness of only +0.03.

Skewness can't tell you how the next round will evolve. For that, you have to phone your 1-900-PSYCHIC. Rather, skewness helps by providing a rational basis for deciding which games to play and bets to make, to induce the types of sessions most apt to meet your personal preferences, constraints, and goals. Unless you think the poet, Sumner A Ingmark, was wrong when he wrote:

Gamblers wise know ere beginning,
How much luck they'll need for winning.

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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.