Put Craps Bets onto a Common Ground before Comparing Them

6 September 2004

By Alan Krigman

Most casino bets are resolved on every round. You drop money on the table or into the slot then win, lose, or push -- completing the coup. Not always so in craps. Some tosses may not yield decisions, but are deemed irrelevant or "no action" rather than pushes. This, although they're equivalent to getting your money back then making the bet again. Alternate bets accordingly need different average numbers of rolls to be resolved. So, a "round" in which a wager is decided may not be of the same duration for everyone at a table, or for every bet made by the same person.

Picture $10 placed on the five or nine. Of the 36 ways the dice can land, four win $14, six cost $10, and 26 have no action. In 36 statistically correct rolls, the bet is settled 10 times. Instead, consider placing the four or 10 for $10. Of the 36 dice outcomes, three win $18, six lose $10, and 27 get no action. In 36 statistically correct rolls, the bet is settled nine times.

Another situation occurs with $5 each on the five and nine, as opposed to $10 on either. Here, 36 statistically correct rolls provide eight ways to win $7 and six to lose $10, so the bet is settled 14 times. It gets messier yet when money is distributed over more boxes. With "$32 across," for instance, 36 statistically correct rolls have six ways to win $9, 18 to win $14, and six to lose $32, so the bet is settled 30 times.

Craps bets could be more rationally compared with each other, or among games, by treating every throw as a trial that wins, loses, or pushes. On this basis, the accompanying table shows expected or average loss due to edge, bankroll swing, and skewness per throw of the dice for every dollar at risk on some representative Place bets. The multi-number bets assume one "unit" at risk on each box (e.g., $5 each on four and 10, $5 each on five and nine, and $6 each on six and eight). For reference, the same statistics are given for jacks-or-better video poker in a game with 8-for-1 and 6-for-1 returns on full houses and flushes, respectively.

Characteristics of various bets on a trial-by-trial basis,
per dollar at risk

bet	expected loss due to edge	average bankroll fluctuation	skewness
Place 4 or 10	$0.0167	$0.661	+1.18
Place 5 or 9	$0.0111	$0.620	+0.63
Place 6 or 8	$0.0046	$0.596	+0.28
Place 4 and 10	$0.0167	$0.549	-0.18
Place 5 and 9	$0.0111	$0.525	-0.56
Place 6 and 8	$0.0046	$0.511	-0.81
Place 4, 5, and 6	$0.0104	$0.491	-1.04
Place 4, 5, 6, 8, 9, 10	$0.0104	$0.451	-1.65
J-B VP 8-6 return	$0.0165	$4.407	+205.99

Casino bets are unlike those on lotteries or sports, however, because they're made repeatedly in sessions rather than singly on events. It's therefore more meaningful to think about where players can anticipate being at the end of a session, than to focus on individual coups. Session projections can be made from the round-by-round values, by factoring in the speed of the game. For craps, a reasonable estimate is roughly 100 throws/hour. For video poker, 600 spins/hour is a good figure. At these rates, someone betting strictly as indicated could be 95 percent certain of finishing an hour-long session between the loss and win limits shown in the second table -- again per dollar at risk per round, so multiply by the actual amount at risk in each case.

Range over which players can expect to finish hour-long sessions,
per dollar at risk

bet	loss	win
Place 4 or 10	$15.53	$11.93
Place 5 or 9	$14.08	$11.68
Place 6 or 8	$12.89	$11.90
Place 4 and 10	$13.21	$9.61
Place 5 and 9	$12.11	$9.71
Place 6 and 8	$11.12	$10.12
Place 4, 5, and 6	$11.32	$9.08
Place 4, 5, 6, 8, 9, 10	$10.50	$8.26
J-B VP 8-6 return	$225.79	$205.99

Offsets between predicted session ups and downs result from the house edge. Differences in profits for similar losses, as in placing four-five-six as opposed to six-eight, follow partly from skewness. For instance, the more negative skew for four-five-six than six-eight suggests that the former gives solid citizens a greater chance to win less. Another reminder that the better you understand what you're doing, the more you're apt to fulfill your personal predilections and preferences. In the casino as well as the real world. Here's how the poet, Sumner A Ingmark, put it:

Those who think life's serendipitous,
Hover o'er an edge precipitous.