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# The most enduring casino games are the most elegant

26 November 2012

In popular parlance, “elegant” is often taken to connote fancy or elaborate. Strictly speaking, and especially when scientists use the term, it means simple in the sense of uncomplicated – and implies praise. Similarly, the most enduring casino games tend to be those that are most elegant. Even craps, which may seem to be complex at first glance, is built around rolls of simple pairs of dice no different than those most solid citizens used in board games they learned as children.

Casino games don’t get more elegant than roulette. Every coup ends with a ball in a compartment of a wheel corresponding to a “spot” on the table layout. Chances are equal for all compartments. So, on a “double zero” wheel with 38 positions, the probability of finishing on any spot is one out of 38. This chance can be expressed equivalently as 1/38, 2.63 percent, or 37-to-1.

What makes roulette interesting is the variety of bets players can make at it. Alternate wagers represent different groups of spots treated as individual propositions. Anything from particular “straight-up” numbers to two-spot splits, three-spot rows or streets, four-spot corners, six-spot double rows or streets, 12-spot columns or “dozens,” or 18-spot “outside” groups. And one or more such sets, exclusive or overlapping, can be bet simultaneously.

Hits on single spots earn 35-to-1 and players recover the money at risk. With one exception on double- and none on single-zero tables, payoffs for other bets follow directly from the straight-up level of 35-to-1. The ratios are as follows.

• Two-spot splits: Bet \$2, effectively \$1 on each of two spots. If one of the spots hits, it pays \$35; the missed spot loses \$1. Net profit is \$35 - \$1 = \$34, which is 17 times the original \$2 wager. Whatever is bet on the split, wins are worth 17-to-1.
• Three-spot rows or streets: Bet \$3, effectively \$1 on each of three spots. If one of the spots hits, it pays \$35; the two missed spots lose \$1 each. Net profit is \$35 - 2 x\$1 = \$33, which is 11 times the original \$3 wager. Whatever is bet on the row, wins are worth 11-to-1.
• Four-spot corners: Bet \$4, effectively \$1 on each of four spots, If one of the spots hits, it pays \$35; the three missed spots lose \$1 each. Net profit is \$35 - 3 x \$1 = \$32, which is 8 times the original \$4 wager. Whatever is bet on the corner, wins are worth 8-to-1.
• Six-spot double rows or streets: Bet \$6, effectively \$1 on each of six spots, If one of the spots hits, it pays \$35; the five missed spots lose \$1 each. Net profit is \$35 - 5 x \$1 = \$30, which is 5 times the original \$6 wager. Whatever is bet on the corner, wins are worth 6-to-1.
• 12-number columns or dozens: Bet \$12, effectively \$1 on each of 12 spots If one of the spots hits, it pays \$35; the 11 missed spots lose \$1 each. Net profit is \$35 - 11 x \$1 = \$24, which is 2 times the original \$12 wager. Whatever is bet on the column or dozen, wins are worth 2-to-1.

This consistency of payoffs leads to another elegant aspect of roulette. Namely, the house edge or advantage is the same for all bets other than the one exception mentioned above. The edge is the theoretical fraction earned by the house as a result of the odds against a win being slightly steeper than the payoff. The value can be calculated as the probability of success multiplied by the payoff, minus the probability of failure multiplied by the bet, all divided by the amount bet. Here’s how it works on double-zero games.

• \$1 on a single spot: [(1/38) x \$35 - (37/38) x \$1]/\$1 = -(\$2/38)/\$1 = - 5.26%
• \$2 on a 2-spot split: [(2/38) x \$34 - (36/38) x \$2]/\$2 = -(\$4/38)/\$2 = -5.26%
• \$3 on a 3-spot street: [(3/38) x \$33 - (35/38) x \$3]/\$3 = -(\$6/38)/\$3 = -5.26%
• \$4 on a 4-spot corner: [(4/38) x \$32 - (34/38) x \$4]/\$4 = -(\$8/38)/\$4 = -5.26%
• \$6 on a 6-spot double street: [(6/38) x \$30 - (32/38) x \$6]/\$6 = -(\$12/38)/\$6 = -5.26%
• \$12 on a 12-spot column: [(12/38) x \$24 - (26/38) x \$12]/\$12 = -(\$24/38)\$/12 = -5.26%

At a single-zero table, the wheel has 37 rather than 38 compartments. Payoffs for the combinations are the same as in double-zero games. Substituting 37 for 38 in the calculations for edge would yield -1/37 or -2.70 percent for all bets

The 35-to-1 single-spot payoffs are part of the elegance of roulette. At any other level, say 36-to-1, payoffs for combinations would be in fractional rather than whole dollars. Otherwise, earnings would differ for the same total at risk on consolidated propositions as the individual spots bet separately. That would be messy, not elegant. It would defy Occam’s Razor – a principle of logic and science first stated in the 14th century and popularized by the poet, Sumner A Ingmark, as:
When complication is propounded,
Alarms and warnings should be sounded.