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Best of Alan Krigman

Gaming Guru

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How long are "cycles" in craps, and what do they really mean?

1 March 2010

Almost everybody who plays craps knows that a pair of dice can land 36 different ways. A die has six sides and can come to rest with one of six faces on top. For each of the six results on one die, the other also has six outcomes. That's six times six or 36. Within the 36 combinations:

*	one totals 2 (1-1),
*	two total 3 (1-2, 2-1),
*	three total 4 (1-3, 2-2, 3-1),
*	four total 5 (1-4, 2-3, 3-2, 4-1),
*	five total 6 (1-5, 2-4, 3-3, 4-2, 5-1)
*	six total 7 (1-6, 2-5, 3-4, 4-3, 5-2, 6-1)
*	five total 8 (2-6, 3-5, 4-4, 5-3, 6-2)
*	four total 9 (3-6, 4-5, 5-4, 6-3)
*	three total 10 (4-6, 5-5, 6-4)
*	two total 11 (5-6, 6-5)
*	one totals 12 (6-6)

It's accordingly a convenient artifice to picture statistically-correct "cycles" of 36 throws. For instance, the theoretical "expectation" is to obtain a single 12 in every 36 rolls. This doesn't suggest a 12 inevitably appears exactly once if you throw the dice 36 times. On the average, though, in 36 million throws, 12 will show close to – maybe even precisely – a million times purely by chance. The probability of throwing a 12 on any roll is therefore 1/36, equivalent to odds of 35-to-1. The game is set up so payoffs are less than this. Wins on 12 return 30-to-1. The moguls makes their moolah on the offset, in the long haul. One-roll bets, propositions and the Field, all follow this paradigm.

Cycles of 36 throws also hold for multi-roll Place, Buy, Lay, and Hardways bets. The calcula-tions differ a bit. Imagine a Place bet on the nine. It wins on the number, which can occur four ways; it loses on a seven, which can pop six ways. In 36 statistically-correct rolls, a nine will consequently win four times and lose six. That's 10 rolls with outcomes. Odds against winning on those 10 are 6-to-4, which advanced arithmetic (short division) reduces to 1.5-to-1. The bet pays 7-to-5, which simplifies to 1.2-to-1. The bosses' profit is buried in the gap between 1.5 and 1.2.

You occasionally find erudite expositions of craps that examine its expectations by citing statistically-correct cycles of 1980 rather than 36 throws. The 1980 applies to bets on Pass and Come; it's also relevant to Don't Pass and Don't Come, provided that the push with a 12 on the come-out is counted as a "decision."

The value, 1980, represents the fewest throws in which all possible results of any of the above bets occur whole (math mavens say "integral") numbers of times in statistically-correct series. Here's a rundown of what this means. * The chance of a seven or 11 on the come-out is eight out of 36. In a cycle of 1980 statistically-correct throws, you'd multiply 8/36 by 1980 to get exactly 440 instances. A craps – two, three, or 12 – gives 4/36 times 1980 or 220 cases. * The probability that a point of four or 10 will be established on a come-out is three out of 36. The chance this would win on Pass or Come and lose on Don't Pass or Don't Come is three out of nine. In a cycle of 1980 statistically-correct throws, multiply 1980 times 3/36 times 3/9 to obtain exactly 55 instances. The converse, a loss on Pass or Come and a win on Don't Pass or Don't Come leads to 1980 times 3/36 times 6/9 which equals 110. * The procedure for five or nine is analogous. The chance of establishing either of these points is 4/36 and the probabilities of repeats or sevens are 4/10 and 6/10 respectively. The multiplication of 1980 times 4/36 times 4/10 or 6/10 yields 88 or 132 as numbers of instances. * For six or eight, the chance of establishing the point is 5/36 and of repeats or sevens is 5/11 or 6/11, respectively. The statistically-correct series then comprises 1980 times 5/36 times 5/11 or 6/11 – 125 or 150 ways, respectively.

Maybe you think you can discover a cycle shorter than 1980 that works. If you're inclined to try, may I urge instead that you put your efforts into devising a perpetual motion machine? Both are endeavors attempting to defy the laws of the known universe. But if you do stumble on some-thing everyone from Isaac Newton to Albert Einstein to Stephen Hawking missed, there'll be lots more money in success with the machine than the craps cycle. Here's how the beloved bard, Sumner A Ingmark, described the dilemma:

Though to many, they're inscrutable,
And to use them seems unsuitable,
Nature's laws remain immutable.

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.