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Who should craps buffs thank or blame for too few or too many sevens?

28 November 2011

Craps aficionados know – or should – that the chance of a seven showing on a toss is one out of six. There’s no great mystery to this probability. It follows from the totals on the faces of two balanced dice. Here are all the possible combinations: 1-1, 1-2, 1-3, 1-4, 1-5, 1-6, 2-1, 2-2, 2-3, 2-4, 2-5, 2-6, 3-1, 3-2, 3-3, 3-4, 3-5, 3-6, 4-1, 4-2, 4-3, 4-4, 4-5, 5-6, 5-1, 5-2, 5-3, 5-4, 5-5, 5-6, 6-1, 6-2, 6-3, 6-4, 6-5, 6-6. Count ‘em; there are 36 ways for the cubes to land. Then count the totals; six combinations add up to seven. That’s six out of 36, which reduces to one out of six.

What if you hurl the hexahedrons six times? How confident can you be that the seven will pop exactly once out of the six? If the six tosses, by happenstance, were statistically correct, you’d get a single seven. But a statistically correct series is an artifact of arithmetic used to illustrate some aspects of the laws of probability; it’s not something that augurs the results of specific situations.

The laws of probability can be employed to find the answer to an arguably more meaningful question. Namely, what are the chances of various numbers of sevens in six throws. The math shows the probabilities to be: zero – 33.4898 percent, one – 40.1878 percent, two – 20.0939 percent, three – 5.3584 percent, four – 0.8038 percent, five – 0.0643 percent, six – 0.0021 percent. A single seven is the most likely case but not by much, and the chance is only about 40.2 percent. Instances of zero sevens aren’t far behind, at roughly 33.5 percent. And two sevens in six throws also has a good likelihood of showing, at 20.1 percent. Chances of three to six sevens drop rapidly, but are still in a range where it shouldn’t be too astonishing to see them regularly.

Make believe, instead of six, you think in terms of 36 rolls. Here, a statistically correct series would have the seven pop six times. Maybe you suspect that the outlook for six sevens in a series of this length would be greater than that of one in six rolls, what with the law of averages making predictions more reliable as sample sizes increase. Sorry. The potentiality for exactly six sevens in 36 rolls is less by far, not more, at 17.587 percent. This, owing to the probabilities being spread over a wider range – zero to 36 rather than zero to six instances.

What about chances of from zero to five sevens – fewer than the average of six – in 36 rolls. The figures are: zero – 0.1411 percent, one – 1.0158 percent, two – 2.5553 percent, three – 8.0587 percent, four – 13.2968 percent, and five – 17.0199 percent. The sum of these fractions, the probability of fewer than six sevens in 36 rolls, is 43.0876 percent – closer to 50-50 than you might have guessed. Similarly for the likelihood of more than six sevens. The chances are: seven – 15.0748 percent, eight – 10.9292 percent, nine – 6.9004 percent, ten – 3.6722 percent, eleven – 1.7360 percent, and for all greater occurrences combined – 1.1126 percent. In all, the likelihood of exceeding six sevens is 39.3252 percent. Less, actually, than that of being under six.

The fact that the average number of sevens in six rolls is 1.0 and in 36 rolls is 6.0 therefore doesn’t imply that lower or higher rates of occurrence are unusual. These other frequencies are not only to be anticipated; they contribute to what makes the average by offsetting one another. You can calculate the average by multiplying each of the instances by its respective probability and adding the products. For instance, with six throws, this would be (0.334898 x 0) + (0.401878 x 1) + (0.200939 x 2) + (0.053584 x 3) + (0.008038 x 4) + (0.000021 x 5) = 1.0.

The averages are meaningful mainly to the casino bean counters, who use them to set the payoffs so the house earns a profit from the games. The joints don’t actually bank the bread on every coup, however, but on the net outcomes of hundreds of thousands or millions of throws.

Say that a million players each undergoes a 36-throw sequence. The number of solid citizens who encounter various numbers of sevens in their 36-throw series would be relatively close to the theoretical probability of the phenomenon for each individual multiplied by the million people.

On this basis, and ignoring the order in which results occur, players making Pass, Come, Place, and Buy bets would be ecstatic having fewer instances of sevens than the average. About 0.1411 percent of the million – 1,411 would have had none in their 36 throws; similarly, 10,158 would have had one, 35,553 would have seen two, 80,587 would have gotten three, and so forth. Others bettors would be anguished by enduring over the average number of the little devils – including about 109,292 who got eight, 68,004 who received nine, 37,772 who saw 10, and so on.

As far as the casino is concerned, happy punters who pile up the pelf would be offset by sadsacks who take a bath. It’s all in the mathematics. But don’t try to tell this to the 1,411 who raked in the riches with no sevens in 36 rolls and imputed the profits to their craps acumen, or to the 1,135 who went to the cleaners with 14 or 15 sevens in rolls of equal duration and laid the losses on everyone within griping distance. For, as that insightful inkster, Sumner A Ingmark, indicated:
Success has many mothers,
While failure’s blamed on others.

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.