Roulette Physics

roulette physicsJust about as many people have studies the physics of a roulette wheel and ball as have tried to beat the wheel with a roulette system. The reward for being able to predict where the ball is going to land in any given spin on a roulette table is just too tempting!

In this section, we take a look at some basic roulette physics to see if this can help us, in any way, to understand the dynamics of the game and thus to gain an edge. Roulette probabilities are fixed. But that hasn't stopped people from trying to predict those outcomes.

There might be some roulette mathematics involved here by the way!
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The first thing to say, is that the roulette wheel is designed to generate outcomes of pure chance. There is no purer form of gambling, and although most wheels are not 100% random (they may be biased, or the dealer may have a signature), they are as close to random as you can possibly get. Casinos aren't bothered about having perfect roulette wheels. They just need them perfect enough so that humans are unable to spot any trends.

Let's take an American Roulette Wheel. There are 38 pockets into which the ball can fall, and all are the same size. The probability of the ball landing in any of them is equal. You could say that a roulette wheel is a random number generator or an RNG.

But, and it's a big but- the result isn't determined by an electronic random number generator like it is in virtual or video roulette. It is determined by the mechanics of a ball going round a wheel, and friction and gravity acting on that ball. Eventually the ball will lose all of its kinetic energy thanks to friction with the wheel and the air, and will eventually bounce across pockets losing more and more energy faster and faster until it comes to a stop.

In theory, if you are able to measure certain parameters, you should be able to work out the pocket into which the ball will fall. Even if you are unable to predict the exact pocket, you should be able to predict a "zone" of numbers. And that is enough in roulette to give you an edge, because of course you can make multiple single number bets.

Visual spotting, or even lasers have been used to collect the necessary initial values of the variables in the system. All this becomes easier if the wheel is biased- even a minor tilt of the rotor, for example, can create shadow zones on the wheel where the ball never falls.

Here we get into the actual physics of a roulette wheel, a topic that has been covered by many scientists including , using the work of Edward Thorp who wrote Elementary Probability (1966), The Mathematics of Gambling (1984) and several mathematical papers on probability, game theory, and functional analysis and Eichberger who has attempted to beat roulette with a computer in his Roulette Physics paper.

In these approximations, friction and air resistance need to be plugged in to the model. Another paper worth looking at, as this comes from the casino's perspective, is Dixon's Roulette Wheel Testing in which he claims that an angle of as little as 0.1° will cause a discernable bias in the wheel.

The Physics of Roulette

Friction and Drag

Let's look at a roulette wheel. It consists of an outside s a rim along which the ball rolls at the beginning of its journey. At some stage  the ball will drop down from the rim when it loses momentum and travel towards the centre of the wheel. The ball will hit a set of bumps, which will send the ball scattering in a chaotic fashion. Then the ball reaches the inner section of the wheel, with 38 identically sized pockets into which it can land.

Say there was no friction, drag, or tilt, the ball would roll around the rim of the wheel in the opposite direction to the wheel spin, infinitely. It's path can be determined by the initial angular velocity of the ball and the initial angular velocity of the wheel. Here we are going to use Eichberger's equation of motion for the wheel without tilt:

roulette physics

ω is the angular velocity of the ball, and α is the angular acceleration of the ball. The constants a and b refer to the effects of friction and drag

Tilted Wheels
If the wheel is tilted, (ie you have a biased wheel), you need additional parameters to describe this. Andy Hall (2007) has written a paper on this called the Forbidden Zones of Roulette Wheels, which make for interesting reading if you are keen on roulette physics. His equation for tilted wheels is as follows:

roulette equation

The ball's angular acceleration α, now depends on the speed of the ball, AND its location, theta. This is due to the tilt- in some areas the ball is deccelerating up the tilt, and in others it is accelerating down it.

Using these and other equations to model the ball's behaviour, the authors have made claimed that they are able to predict the final resting place of the ball with a high enough degree of accuracy to be able to get an edge over the casinos, by predicting:

Where the ball leaves the Rim and 
Working out the Departure Angle of the Ball

The amount of tilt that a wheel has affects how big the "shadow zones" are on roulette wheels, as modelled by these equations. But importantly, these shadow zones or "forbidden zones" relate to where the ball comes off the rim of the outer wheel, not where it stops. The casinos still have one ace up their sleeve- and that is the "bumps" that chaotically scatter the ball in all directions.

This is a far harder thing to model. Can you beat roulette with chaos theory? Well, that's a whole different subject!