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Best of Alan Krigman
Level the Playing Field to Compare Craps Bets27 September 1999
It's no secret that casinos have different edges on the various bets they offer. This is a common gauge for comparing alternate wagers. While edge has relatively minor impact on sessions of reasonable length, high values give casinos the most profit so -- other factors being equal -- low values are best for players.
Technically, edge is the percentage of each wager the house expects to earn, based on the chances players will win and what they're paid when they do. With a few exceptions, casinos don't actually collect commissions on every round of play. But, over an extended period, their net revenues home in on the amount predicted by multiplying edge times the total bet or "handle."
In most instances, edge is figured on every bet on every round -- win, lose, or push. So bets can be directly compared with one another. For instance, the edge betting on individual numbers or on 12-number columns at 00 roulette is 5.26 percent. From the perspective of house advantage alone, these bets are equivalent.
In craps, however, edge is stated not in terms of numbers of rounds, but of decisions. And different bets have unique resolution cycles, so direct comparisons can be misleading.
Assume you place the four for $5. This bet is decided only if the dice show a four and you win $9, or a seven and you lose $5. The dice can show four three ways and add up to seven six ways. The resolution cycle comprises three plus six or nine rolls. And the house's theoretical profit is 6x$5 - 3x$9 or $3 on the 9x$5 or $45 total bet. The edge is $3 divided by $45, which equals 6.67 percent of the money you risk. Ditto for place bets on the 10.
Instead, make believe you place the five for $5. Now you'll win $7 when the dice show five and lose $5 when the seven appears. The resolution cycle comprises 10 rolls -- four ways to win, six to lose. So the house's theoretical profit is 6x$5 - 4x$7 or $2 on the 10x$5 or $50 total bet. The edge is $2 divided by $50 or 4 percent. Likewise for bets on the nine.
Similar reasoning holds for place bets on the six or eight. On a $6 bet, you have five ways to win $7 and six ways to lose $6. The theoretical casino profit is 6x$6 - 5x$7 or $1 on 11x$6 or $66 total bet. The edge is $1 divided by $66 or 1.515 percent.
Some gambling gurus argue that craps bets should be compared based on rolls and dollars rather than decisions and percentages. They rationalize that bettors usually care more about what happens in sessions of specified lengths than during hypothetical resolution cycles. Since, in a given interval, players expect the most decisions on sixes and eights and the fewest on fours and 10s, relative theoretical casino rakes per roll or per unit time aren't the same as they are per decision. Further, edge based on resolution cycles loses meaning when bets are made on several numbers, especially with different amounts on each.
The per-roll approach uses the fact that the theoretical casino profit in 36 statistically-correct throws and in the matching resolution cycle are the same for equal bets. This is $3/5 or $0.60 per dollar on the four or 10, $2/5 or $0.40 per dollar on the five or nine, and $1/6 or $0.1667 per dollar on the six or eight. Divide each figure by 36 to get per-roll rakes of 1.67 cents per dollar bet on the four or 10, 1.11 cents per dollar on the five or nine, and 0.46 cents per dollar on the six or eight.
Here's how comparisons would be made with these values. Assume you bet $30. With it all on six or eight, the house expects to earn $0.0046x30 or $0.13 per roll. When it's all on five or nine, the casino expects $0.0111x30=$0.33 per roll. All on four or 10 nets the house a theoretical $0.0167x30=$0.50 per roll. Likewise, for multiple numbers, $10 on four and $20 on five are worth $0.0167x10 + $0.0111x20 = $0.39 per roll to the casino.
Presented with this way to standardize evaluations of disparate craps bets, the poet Sumner A Ingmark reflected:
Of objectivity 'bout worth?
Best of Alan Krigman