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Best of The Midnight Skulker

Gaming Guru

 

To Continue, or Not to Continue: That Is the Question

27 April 2001

The mathematics of the game of craps is based on two assumptions:

  1. The rolls are independent.
  2. The results of rolls are uniformly distributed across the 36 possible combinations of two six-sided dice (a distribution I call "The Perfect 36").

Proceeding from these assumptions, mathematics demonstrates that the player is doomed.

OK, everybody calm down. Mathematics does not say every player is doomed, it says that players taken as a whole are almost certainly doomed. Individual players can and do show a profit for a weekend, some are in the black over several weekends, and I presume there are even a very few individuals who are or have been in the black for a lifetime, despite frequent trips to the tables. These anomalies do not violate the numbers, which merely say that the longer one plays, the less likely it is that one will be ahead. As penned by Sumner A Ingmark, prefered poet of Alan Krigman:

The statisticians don't know how,
To buy stocks that will beat the Dow,
Or win at gambling here and now.
The weakness is their math depends,
On testing time that never ends.

Only in the infinite is it impossible to come out ahead.

One way of "winning at gambling here and now" is to take advantage of short-term trends. But how? If one detects a trend, does one (with apologies to The Bard) bet it
To continue, or not to continue, that is the question.
Whether 'tis more rewarding to risk
The slings and arrows of reversing fortune,
Or to make bets against the trending seas,
And by opposing, profit from them?

Those who bet that a trend will reverse cite the Law of Large Numbers, which in layman's terms says that the distribution of actual results of dice rolls will get closer and closer to the theoretical distribution as the number of trials is increased. Hence, the argument goes, if a particular result occurs much more or less frequently than predicted by probability there must be a correction in the opposite direction for the Law of Large Numbers to hold.

The above logic is an example of what is known as the Gambler's Fallacy. A web search for the keywords "Gambler's Fallacy" returned multiple hits when I tried it, so I will say only that betting a trend to reverse puts one in a mathematical no-man's-land. The logic accepts the part of the mathematical model that says the distribution of results will approach the theoretical distribution over time, but rejects the part that says the probability of an event occurring remains constant in a game of independent trials. Unfortunately, mathematics is a package deal. You accept or reject all of it; you can't pick and choose parts that suit your fancy.

"Wait a minute!" you were about (or wish you had been about) to say. "Doesn't the math say trends have no predictive value?" You refer (or wish you had refered) to "the dice have no memory" mantra? Yes, that is what the math says. "Then why are we even discussing trends?" you ask (or--well, never mind). Answer: because that is not exactly what the the math says. (Ya like that footwork?)

What the math does say is that trends have no predictive value in a game of random and independent trials (See the base assumptions above.), with "random" taken as implying The Perfect 36. For the math to apply to the game of craps, it must be a game of random and independent trials.

Is it? Over 90 years of modern bank craps experience certainly supports the position that it is. Is it always? Ah, that extra word "always" may leave some room for doubt.

What if there are times when The Perfect 36 is not the distribution in effect? Would, say, a total of an hour or two per week when a different distribution applied produce enough variance in a casino's revenue from craps to draw any attention? I should think not, if for no other reason than that whatever alternate distribution is in effect when The Perfect 36 is not could just as easily favor the Don't side as the Do, and since the majority of players play exclusively on the front side these times of alternate distributions would tend to cancel out.

What might cause The Perfect 36 to be violated? Biased dice, certainly, but the chances of finding any in today's casinos are slim to none. Of the legal possibilities, I consider rhythmic rollers to be the strongest candidate, though I am compelled to note that the only evidence that they exist is anecdotal. Other possibilities range from the ridiculous to the less ridiculous but, except for strengthening the argument that the dice sometimes follow an alternate distribution, what causes that behavior is irrelevant. What is relevant is that for a trend to have predictive value a distribution other than The Perfect 36 must be in effect, and the -- er -- trendy player must hope that the trend being observed is an indicator that such is the case.

Trends occur. Anyone who has spent even a few hours at the craps table has encountered them. Unfortunately, even if some trends are indicators of player-favorable conditions--and make no mistake, I have not come even close to showing that any of them are--other trends will also occur from the normal operation of randomness. Caution and skepticism are therefore advised. But if you do accept the notion that some trends have predictive value, then logically you should bet them to continue.

The Midnight Skulker
The Midnight Skulker has been playing craps for over three decades and has played almost everywhere in the country. He is a computer expert and a frequent contributor to Internet newgroups, where his opinions and observations have earned him much respect.
The Midnight Skulker
The Midnight Skulker has been playing craps for over three decades and has played almost everywhere in the country. He is a computer expert and a frequent contributor to Internet newgroups, where his opinions and observations have earned him much respect.