Testing for dice control at craps

Abstract

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 $\theta\in[0,1]$, 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 $H_0:\theta = 0$ (no control) versus $H_1:\theta > 0$ (some control). We also want to test $H_0:\theta\le\theta_0$ versus $H_1:\theta>\theta_0$, where $\theta_0$ is the "break-even point." For the tests considered we estimate the power, either by normal approximation or by simulation.