The European roulette wheel. In the left panel, one can see a portion of the rotating roulette wheel and surrounding fixed track. The ball has come to rest in the (green) 0 pocket. Although the motion of the wheel and the ball (in the outer track) are simple and linear, one can see the addition of several metal deflectors on the stator (that is, the fixed frame on which the rotating wheel sits). The sharp frets between pockets also introduce strong nonlinearity as the ball slows and bounces between pockets. The panel on the right depicts the arrangement of the number 0 to 36 and the coloring red (lighter) and black (darker).
The dynamic model of ball and wheel. On the left, we show a top view of the roulette wheel (shaded region) and the stator (outer circles). The ball is moving on the stator with instantaneous position while the wheel is rotating with angular velocity (note that the direction of the arrows here are for illustration only, the analysis in the text assume the same convention, clockwise positive, for both ball and wheel). The deflectors on the stator are modelled as a circle, concentric with the wheel, of radius . On the right, we show a cross section and examination of the forces acting on the ball in the incline plane of the stator. The angle is the incline of the stator, m is themass of the ball, is the radial acceleration of the ball, and g is gravity.
The case of the crooked table. The blue curve denotes the stability criterion (6), while the red solid line is the (approximate) trajectory of the ball with indicating two successive times of complete revolutions. The point at which the ball leaves the rim will therefore be the first intersection of this stability criterion and the trajectory. This will necessarily be in the region to the left of the point at which the ball's trajectory is tangent to Eq. (6), and this is highlighted in the figure as a green solid. Typically a crooked table will only be slightly crooked and hence this region will be close to but biased toward the approaching ball. The width of that region depends on , which in turn can be determined from Eq. (6).
Hand-measurement of ball and wheel velocity for prediction. From two spins of the wheel, and 20 successive spins of the ball we logged the time (in seconds) T(i) for successive passes past a given point (T(i) against T(i + 1)). The measurements T(i) and T(i + 1) are the timings of successive revolutions—direct measurements of the angular velocity observed over one complete rotation. To provide the simplest and most direct indication that handheld measurements of this quantity are accurate, we indicate in this figure a deterministic relationship between these quantities. From this relationship, one can determine the angular deceleration. The red (slightly higher) points depict these times for the wheel, the blue (lower) points are for the ball. A single trial of both ball and wheel is randomly highlighted with crosses (superimposed). The inset is an enlargement of the detail in the lower left corner. Both the noise and the determinism of this method are evident. In particular, the wheel velocity is relatively easy to calculate and decays slowly, in contrast the ball decays faster and is more difficult to measure.
Predicting roulette. The plot depicts the results of 700 trials of our automated image recognition software used to predict the outcome of independent spins of a roulette wheel. What we plot here is a histogram in polar coordinates of the difference between the predicted and the actual outcome (the “Target” location, at the 12 o'clock position in this diagram, indicating that the prediction was correct). The length of each of the 37 black bars denote the frequency with which predicted and actual outcome differed by exactly the corresponding angle. Dotted, dotted-dashed, and solid (red) lines depict the corresponding 99.9%, 99%, and 90% confidence intervals using the corresponding two-tailed binomial distribution. Motion forward (i.e., ball continues to move in the same direction) is clockwise, motion backwards is anti-clockwise. From the 37 possible results, there are 2 instances outside the 99% confidence interval. There are 7 instances outside the 90% confidence interval.
Parameter uncertainty. We explore the effect of error in the model parameters on the outcome by varying the three physical parameters of the wheel (a) and introducing uncertainty in the measurement of timing events used to obtain estimates of velocity and deceleration (b). In (a), we depict the effect of perturbing the estimated valuesof (green—affine, increasing steeply) (red—affine, decreasing) and (blue—affine and increasing slowly) from 90% to 110% of the true value. In (b), we add Gaussian noise of magnitude between 0.5% and 10% the variance of the true measurements to initial estimates of all positions and velocities. Horizontal dotted lines in both plots depict error corresponding to one whole pocket in the wheel. The vertical axis is in radians and covers —half the wheel. In the upper panel, least variation in outcome is observed with errors in estimation of .
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