Dancing Between Raindrops

In most of the hundreds of games involving dice, the rules attach no special significance to any of the possible totals. For example, the board games Backgammon and Monopoly give special status to no numbers other than those formed by pairs. In Craps, however, the number Seven is very special. Its critical role in every one of the major games of Craps causes the behavior of Seven to be a fruitful topic of inquiry. Knowing the behavioral tendency of Seven can be very useful for shaping your strategies. If like most players you play on the bright side with bets on Pass and Come, taking Odds and occasionally making Place bets, then Seven is almost completely your enemy. You may want to know how to evade it. Successful evasions are like dancing between raindrops.

Over the long run of infinite rolls, the number Seven will constitute 6/36 of all outcomes. That frequency of Seven is not as interesting as the timing and clustering of its occurrences. In my earlier post labeled “Seven on Dice Road” I suggested that you roll two dice a few hundred times and record the appearance of Seven. Please use your saved record of those rolls to confirm for yourself what I now will relate.

When you are playing Craps, the only Seven that matters is the next one. When will it occur? No one can answer that question with certainty. However, we can answer it with useful probability. Looking forward along Dice Road, we can say when the next Seven becomes probable. For the very next roll, Seven is unlikely. Within the next twenty rolls, however, Seven is extremely likely. What is the future roll at which the next Seven crosses over from unlikely to likely? In other words, after how many future rolls will the cumulative probability of its occurrence exceed 50 percent? The answer is roll number 3.8, which is approximately the fourth roll.

Here is my bold claim: Of all future Sevens, a bit more than half of them will require only four or fewer rolls to occur. Therefore, a bit less than half of them will require five or more rolls to occur. Because an algebraic proof may be opaque and therefore unpersuasive, I invite you to confirm this result empirically for yourself by actually rolling dice, recording the outcomes, and examining the results. You will need to roll at least sixty Sevens for a minimum sample size. For the sake of simplicity, let’s divide the results into Short Sevens and Long Sevens. Even in your small sample you will find that they roughly balance. In the infinite long run they will balance nicely.

Although on average Seven occurs every six rolls, a Seven that stays away until the sixth roll is a Long Seven and is emphatically not average. Considering Dice Road at any starting time, only four future rolls are likely to be adequate to produce the next Seven. You can win a few tavern bets with this one. To be very safe, take five or under.

You may also want to toss a coin to familiarize yourself with the clustering tendencies of Heads and Tails, because Short Sevens and Long Sevens behave very nearly the same way. Over infinite trials, precisely half of all Heads and Tails outcomes will occur in clusters of size three or larger. Moreover, 31 percent will occur in clusters size four or larger. Short Sevens and Long Sevens behave very nearly the same way, which has large implications in Craps. If you play on the bright side, then a cluster of Short Sevens is bad for business. Such a cluster usually results in a “cold” table.

I give the label “Seven Storm” to clusters of very Short Sevens which have three or more Sevens occurring either consecutively or with only one roll separating them. Although consecutive Sevens means one or two Naturals, Short Sevens in general tend to be bad for bright side play and especially bad for Odds and Place bets.

Fortunately for bright side players, roughly half of all Long Sevens will occur in clusters. Especially if a few of them occur as Naturals, a cluster of Long Sevens is the best way of describing a “hot” table and will occupy a long stretch of Dice Road. Only within the span of a Long Seven is there enough space in which to really dance between the raindrops by making Place bets. Long Sevens also allow enough time for points to repeat and Odds bets to win. Most interesting for Craps is the fact that Long Sevens can be very long indeed.


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