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Count the Ways of Craps, and You'll Understand the Whys

13 April 2005

Pairs of dice can land 36 ways. Some players find this puzzling, since rolls such as 4-2 and 2-4 are indistinguishable yet count as two ways. To understand it, assume for the sake of argument that the dice are of different colors, red and green. Say red shows a one. Green can then be anything from one to six. That's six ways right there. Now pretend red lands with two-up. Green can again be anything from one to six. That's six more ways. In all, there are six ways red can finish, each accommodating six green results. The total is six times six or 36.

You can figure how many ways each number can be formed. A four, for example. Red can show a one and green a three, red a two and green a two, red a three and green a one ?? three ways. The chance of throwing a four is therefore three ways out of the 36 possible combinations. This is three out of 36, which by short division is one out of 12 and long division 8.33 percent.

What about a seven? Red can show a one and green a six, red a two green a five, red a three green a four, red a four green a three, red a five green a two, and red a six green a one ?? six ways in all. So the chance is six out of 36; or, invoking the advanced math of grade school arithmetic, one out of six or 16.7 percent.

Craps is built around the prospects of various numbers hitting, or for "hardways" and "hops" bets, forming. For any wagers, payoffs are multiples somewhat shy of the odds overcome to win. The margin between the two gives the house its edge.

To see how this works, consider a Place bet on the nine. The bet wins if a nine pops and loses on a seven; nothing else matters. A nine can be made four ways (3-6, 4-5, 5-4, and 6-3); the seven, as noted, six ways. So, the odds against winning are 6-to-4 or (short division again) 1.5-to-1. Payoffs are 7-to-5 or 1.4-to-1. The joints get their juice on the offset between 1.5 and 1.4.

What would be needed to eliminate the edge in craps or make it a "positive expectation" game? The answer depends on the specific wager at issue, since it differs for the available alternatives.

On the nine, for instance, these outcomes could be achieved by modifying the payoff, the chances of throwing a nine, or the likelihood of a seven. You can get an idea of the required changes by taking each of these elements individually.

The game would be "fair" were a win paid 1.5-to-1, \$7.50 instead of \$7 for every \$5 bet, holding the chances of nines and sevens at their usual values. Keeping the 7-to-5 payoff, the game would be fair were the odds against winning also 7-to-5. This implies introducing means by which the shot at a nine rose to 4.3 from four out of 36, 11.9 rather than the usual 11.1 percent, with the seven remaining at 16.7 percent. Alternately, the bias might lower the probability of a seven to 15.6 from the normal 16.7 percent, with the nine staying at 11.1 percent. Going further in the same directions, higher probabilities of nines or lower of sevens would shift the advantage from the bosses to the bettors.

The various Place bets can be compared by finding how far the probability of a seven would have to drop for edge to be zero, with everything else kept standard. For fours and 10s, the potential for a seven would have to fall to 15 percent. For fives and nines, the decrease would have to be 15.6 percent. And for sixes and eights, the reduction would have to be to 16.2 percent.

Certain solid citizens think dice can be set and thrown to lessen the likelihood of a seven. Others doubt such a thing, but realize that sevens may be more or less frequent than the statistical average in typical sessions, and hope to be at the rail for the latter. A few may read this and infer that counting sevens for a while can suggest when to enter a game. Perhaps because a defect in the dice make sevens scarcer than expected. Or because there have already been too many sevens so the law of averages will hold them back to restore the balance of nature. Both embody the logical lapse lamented in this laconic lyric by Sumner A Ingmark:

It's easy to foster fallacious delusions,
Just fiddle the facts to fit foregone conclusions.

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Best of Alan Krigman
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.