Testing for dice control at craps

Dice control involves “setting” the dice and then throwing them carefully, in the hope of influencing the outcomes and gaining an advantage at craps. How does one test for this ability? To specify the alternative hypothesis, we need a statistical model of dice control. Two have been suggested in the gambling literature, namely the Smith-Scott model and the Wong-Shackleford model. Both models are parameterized by $θ\in[0,1]$, which measures the shooter’s level of control. We propose and compare four test statistics: (a) the sample proportion of 7s; (b) the sample proportion of pass-line wins; (c) the sample mean of hand-length observations; and (d) the likelihood ratio statistic for a hand-length sample. We want to test $H_0:θ= 0$ (no control) versus $H_1:θ> 0$ (some control). We also want to test $H_0:θ\leθ_0$ versus $H_1:θ>θ_0$, where $θ_0$ is the “break-even point.” For the tests considered we estimate the power, either by normal approximation or by simulation.

💡 Research Summary

The paper addresses the long‑standing claim that a skilled shooter can “control” dice in the casino game of craps, thereby reducing the probability of undesirable outcomes (especially the roll of a 7) and gaining a positive expected return on a pass‑line bet. To test such a claim, the author first reviews the mechanics of craps, defining the “hand” (the sequence of rolls from the come‑out until the shooter sevens‑out) and the relevant performance measures: the proportion of 7s, the proportion of winning pass‑line bets, and the hand length.

Two statistical models of dice control that have appeared in the gambling literature are examined. The Smith–Scott model (2018) assumes that each die can be “set” to avoid two opposite faces, leading to three possible die‑sets (A, B, C). The original formulation mixes the single‑die distributions before convolution, implicitly assuming independence between the two dice even when control is present. The author argues that this independence is physically implausible and proposes a modification: the distribution of totals is a convex combination of the fair‑dice total distribution (p_{SS}) and the total distribution obtained when both dice are perfectly set (e.g., (p_{AA}, p_{AB}), etc.). This “mixture of convolutions” respects the fact that the two dice are jointly controlled.

The second model, the Wong–Shackleford model (2005, 2023), is based on the orientation of the dice after a throw. A throw is classified by its “pitch”: zero pitch (both dice rotate the same amount), single pitch (90° difference), or double pitch (180° difference). The skill factor (\theta) is defined as the proportion of double pitches that are converted into zero pitches by a skilled shooter. The joint distribution of the two dice is expressed with indicator functions that increase the probability of zero‑pitch outcomes while leaving single‑pitch probabilities unchanged.

Both models use a single parameter (\theta\in