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What Shift in Instances of Sevens Gets You the Edge at Craps?

3 February 2004

Craps hinges on the seven. Except on one-roll and "hardways" bets, this is the number most desired or dreaded. And when money's on Pass or Come, players first desire then dread it; conversely for those who prefer Don't Pass or Don't Come.

In an unbiased game, the seven is expected to occur on the average of once in every six throws. This because two cubes, each with faces labeled from one to six, can land in 36 different ways, six of which will total seven (1-6, 2-5, 3-4, 4-3, 5-2, and 6-1). And everyone who survived short division in the third grade knows that 6/36 is the same as 1/6. Other potential totals involve fewer combinations. For instance an eight can be formed in five ways (2-6, 3-5, 4-4, 5-3, and 6-2), a five in four (1-4, 2-3, 3-2, and 4-1), an 11 in two (5-6 and 6-5), and so forth.

"Expected" and "average" are statistical terms. They don't mean that a seven will pop exactly once in every six throws, or six times in every 36. To winners' delight and losers' distress, cases of seven in small sets of random trials may be below or above the projected value. Further, "small" in a statistical sense can mean more rolls of the dice than might be experienced in a session, casino visit, or even year of action for reasonably regular players. Add to this the possibility of shooters exerting a degree of control over results by the way the dice are aligned, aimed, and set into motion. Such factors lead to considerations of how different from the expected frequency would the seven have to average during a given period for punters to have an edge.

This isn't as straightforward a question as it may seem. The answer depends on the bet being made. Further, for Pass and Come, there's a trade-off in that fewer sevens hurt during the come-out then help on the point roll, and conversely for Don't Pass and Don't Come. In terms of wins and losses, the order in which various totals appear is also critical. And, if the seven is to rear its head more or less often than six out of 36, other totals have to account for the difference. That is, say you monitor 35 rolls and count five sevens with all else materializing according to the nominal probabilities; if the 36th roll isn't a seven, one of the other numbers will go over the top. In actuality, any 36 outcomes will be spread unpredictably among all alternatives. Even 36 million throws won't hit exactly 6 million sevens, 5 million sixes and eights, and on along the line.

It's still instructive, however, to ignore everything but changes in the proportion of sevens and find the frequency that would give solid citizens a theoretical advantage in the game. This affords a good intuitive grasp on how volatile the game or lucky the gamblers must be to gain the upper hand at craps.

The accompanying table gives the number of sevens in 36 throws, all other factors being disregarded, for the various Place as well as flat Pass and Come wagers to have no edge for either the bosses or the bettors. Fewer sevens than indicated tilt the game toward the players; more increasingly favor the casino.

 bet payoff no of sevens in 36 throws Place four or 10 9-to-5 5.400 Place five or nine 7-to-5 5.600 Place six or eight 7-to-6 5.833 Flat Pass or Come 1-to-1 5.496

The data reveal that modest reductions from the "correct" distribution of sevens suffice to shift the edge to the bettors. For example, with Place bets on the six or eight, a change from the nominal six to 5.833 out of 36 would do it. Flat pass or come bets become "fair" if sevens average 5.496 out of 36; instances don't have to drop as far if Odds are taken on these wagers.

If you believe you can own the hexahedrons by hurling deftly, thinking nobly, living morally, and beseeching the proper gods, these figures show the extent to which you must do so. Of course, a step of the same magnitude in the other direction would boost the edge for the house considerably. So, if you're just depending on being at the right table at the right time, you can see how easily the bottom could fall out. The John Donne of the dice, Sumner A Ingmark, captured this concept with the couplet:

When intangibles you're trusting,