Formula and Methodology Request re: Shannon, Thorp et al.

[gtatix]

I have done a fair bit of research and reading on the subject I am asking about. There is not much - in fact virtually no - detailed information anywhere on the internet about this.

If you have heard of Claude Shannon, Edward O Thorp, Norman Packard, Doyne Farmer and Thomas Bass, then you will know exactly what I am talking about.

Using classical physics and perhaps some chaos formulae, I would like to know how one would go about creating formulas to calculate the probable location of a spinning ball around and then into a bowl. Roulette comes to mind - in fact it is ON my mind.

I know that we would use classic physics formulas for the first part of the problem which is basically a 2-dimensional model. The ball is going around a circle of a specific circumference. The ball continues on this path - decelerating due to friction - and ultimately STARTS to fall inwards.

At this point, we are now dealing with a third dimension as the ball rolls downwards at an acceleration dependant on the slope of the wall and the initial velocity of the ball where it left the rim. It is also still moving slightly forward in the direction it was on before it started to fall. Vectors come to mind here.

Now we could likely be pretty happy with predicting where the ball would hit the wheel at the bottom - but it aint that easy - of course!

The ball is entering a moving wheel at the bottom that has its own circular 2-dimensional motion. For arguments sake - and to make things a little easier - HA - we can assume that this wheel is moving at a constant velocity - no acceleration.

The ball is also LIKELY - but not necessarily - going to hit an obstacle on the way down. There are small bumps around the wheel that are placed relatively uniformly - I forget the name of them as I type - but they have a name. Clearly the name is not important but perhaps the shape is?!

There is obviously a chaos factor and perhaps we don't need to incorporate any extreme formulae to come up with a solution. We don't need to know EXACTLY where the ball should land, we only need to know a relatively general area that would include anywhere from .125 the circumference of the moving wheel at the bottom to perhaps .25 the circumference. And we would only need to be correct a percentage of the time - not 100% but as close to this as possible. I don't know specifically how accurate but let's say 30% of the time.

What I am asking is for as much formulatic information that may be necessary. Definitely some classical physics and likely some chaos and perhaps some probability.

We could also want to look at placing data into a neural network application to come up with some good predictions based on small to medium data.

All input would be greatly appreciated. I will include below some formulae I have dug up. It may have nothing to do with this or it may be wrong.


Y axis¦N1¦*COS(a)-(mg)*COS(a)=0

X axis¦N2¦+¦N1¦*SIN(a)+¦mg¦*sin(@)*COS(Y)=m*¦@ centre)=m*V^2/R=m*[Y')^2]*R

V=Linear Velocity
R=Ball Track Radius
@=Centripedal acceleration

Z axis¦Ffr¦+¦Air Drag¦=m*¦@tan¦=m*Y''*R

Friction Force a This is negative as it is opposing the Z axis

Air Drag is the force that is equal too:

¦Air Drag¦= - 0.5*CD*P*TT*r^2*V^2 (TT is pie) this is also a minus value!

CD is Drag Coefficiaent
P is AIr density
r is the balls Radius

Z axis is always tangentially directed.

------------------------------------------

After some very simple Algerbraic Transformation and incorporating the above formulas we get the next differential equation:

Y''=(a+air*R)*(Y')^2=b*SIN(Y)+c*COS(Y)+d (*)

Where

a is the determining friction factor(Ie 0.004)

Air =-[0.5*CD*P*TT*r^2*V^2]/m

b=a*g*SIN(@)/R

c=b/a

d=a.g.COS(@)SIN(a)+1)/(R.COS(@)

The ball movement sters to this equation only till the moment when it loses the contact with the vertical side of the ball track or:

[N2]=0

So the Drop off condition is:

[(Y')^2]*R+g*COS(@)*tg(a)-g*SIN(@)*COS(Y)=0 (**)

Now let's introduce some real values into the equations and see the predicted results:

TT=3.14
g=9.807
R=0.4
a=16.7.TT/180 inner slope of stator
CD=0.47
r=0.5.21.10^-3 Radius of ball
P=1.22
m=9.10^-3 Mass of Ball
(a)= 0.004 Friction factor for rolling between the ball and the track
@=0.8 grad Tilt Angle

t0= 0 sec
t1=30 seconds

These values determin the time interval of 30 sec since the start of spinning!

I also calcualted the time the ball loses contact with the vertical wall of the ball track, this is when(**) becomes true!

Time till drop off is 17.04 seconds

By this time the ball passes 4935 Grad or 13.7 revolutions from start point!

At this moment the ball has a velocity of 2.7 Rads/Sec or 0.43 Revs per/Sec!



Any and all help appreciated
Cheers

 

[Civilized]

The formula you want not only does not exist, but it cannot exist. Chaos theoy tells us that the game of roulette is sensitive to the smallest details of the setup. The slightest change in the position or direction with which the ball is cast will change the outcome entirely.

The imperfections of a particular roulette wheel might cause it to be biased. The best you could ever do is to simulate a statistically signficant number of rounds on a particular real physical wheel in the hopes of finding flaws with that wheel that would make some outomes more likely.

If you want to apply physics and math in the casino, choose a game that involves strategy as well as chance.

 

[gtatix]

The formula you want not only does not exist, but it cannot exist. Chaos theoy tells us that the game of roulette is sensitive to the smallest details of the setup. The slightest change in the position or direction with which the ball is cast will change the outcome entirely.

The imperfections of a particular roulette wheel might cause it to be biased. The best you could ever do is to simulate a statistically signficant number of rounds on a particular real physical wheel in the hopes of finding flaws with that wheel that would make some outomes more likely.

If you want to apply physics and math in the casino, choose a game that involves strategy as well as chance.

There may be no application for chaos here but you are wrong about predicting AN outcome that would be acceptable and profitable.

You need to do some research on the people I mentioned in the beginning. This HAS been done and gave a measured - scientifically - result that gave the players a 44% edge.

Likely, basic classical physics formulae would suffice with an error factor added and results vs predictions over a relatively short time would graph the prediction vs results well enough to tweak a program or result to give a PROBABLE AREA of the wheel where the ball will fall.

 

[gtatix]

http://nowscape.com/blk/roul/Thorp_First_Wearable_Computer.pdf

 

[Civilized]

That is very surprising, but I guess I stand corrected. This system must be less chaotic then I thought.

Now that you have determined where the ball will strike the wheel, we need to know how much impulse the wheel will impart to the ball.

 

[gtatix]

That is very surprising, but I guess I stand corrected. This system must be less chaotic then I thought.

Now that you have determined where the ball will strike the wheel, we need to know how much impulse the wheel will impart to the ball.

Well, there was a very intelligent man involved - Prof Claude Shannon.
I have thought about this for quite some time. The formulae used would have started out with the basics. If anyone can help - I have forgotten most of my physics whereas I got excellent marks in high school - I would like some direction for starters and then we could move on from there.

Basic formulas needed - and the actual formulas would be great - for centripetal or circular motion. I would assume a coefficient of friction may be needed or perhaps we only need to incorporate time and distance which can be calculated by observation. Once we have time and distance we should be able to calculate (-)acceleration.

Maybe we only need to apply a list of times from observation if we know the distance is a constant.

The predicted falling point of the ball is essential. The orientation of the wheel at this time - what number is at the fall point. The velocity of the wheel - changing over time but constant PER event. The DIRECTION of the wheel - CCW or CW -

Then, the distance from the fall point to the wheel which is likely a vector based on the - still forward - velocity of the ball and the acceleration downward of the ball due to gravity AND the angle of the fall - the angle of the side of the bowl.

Once we know this, we can exclude the fact that the ball will bounce differently virtually ALL of the time! But we don't really worry about this because we don't have to know EXACLTY where it will rest AND we don't need to know ALL THE TIME.

If we can get a general area where the ball lands at different velocities - wheel and ball - and while the wheel is going CCW or CW - the ball is always going CCW - AND I would predict that a more accurate result would be given when the ball is going the direction of the wheel?!

I will stop with the suggestions now but I want it known that chaos is likely NOT necessary. Basic physics should suffice. Using trig because of vectors is needed but that should be about it.

 

[gtatix]

The formulae/formulas in my original post are not mine. I did not apply the numbers or the formulae. IF they are correct, that is great. Does anyone know - first - if they look correct for determining the fall-off point from the launch track and the fall formula down the side of the assembly?

I have no idea if the dimensions are correct - i.e. radius of rim/track, diameter of ball, mass of ball, coefficients etc.

Basically I know nothing! Except that the application seems feasible and has been done.

 

[Civilized]

In the PDF I read the section about 'how they did it' in detail. Based on my experience and that section, they did much more than combine basic formulas. As I suggested they simulated a large number of runs using a real roulette wheel, and in the section is says that they used a strobe lamp and a camera to capture the motion to study it in detail.

The way I would do something like this (I am a physicist, an engineer would do better, I believe Shannon was trained as an electrical engineer) would be a combination of statistical analysis together with looking at what actually happens on a real roulette wheel.

Building a theoretical model, as your trying to do, is helpful if we want to understand the qualitative behavior, but for quantitative statistics about this system we would need to study real roulette throws on a video.

By the way, the computerized video capture software that we have these days would make the job much easier, but I suspect it would still take at least a few weeks of effort to make a useable prediction computer.

 

[gtatix]

By the way, the computerized video capture software that we have these days would make the job much easier, but I suspect it would still take at least a few weeks of effort to make a useable prediction computer.

So, when would you like to get started

I believe - I'd have to reread it - that they did all the tests and shots to verify the software and device.

Really, there are a number of things that can change any result from spin to spin and more likely over a short time. Oil on a croupier's fingers - just skin oil for that matter - the air temperature, atmospheric pressure, the type of material the "wood" of the wheel is made of. But more specifically I think that the chaos that occurs after the ball falls and hits the obstacles around the edge, hits the pockets at a great different number of locations and angles and velocities etc. etc. etc. make it look impossible.

I get the impression that the basic formulae involved are the key to calculating the drop point. This point - where the ball leaves the outer rim and begins to fall - would be the point at which the initial "key number" on the wheel below should be noted.

Once dozens or hundreds of spins are timed and the key formula numbers accounted for - ball deceleration, forward velocity at the start of the fall, wheel direction and wheel velocity and perhaps a couple of others FOR EACH SPIN - I believe the actual manual recording and then data entry of the ultimate pocket position the ball lands in for each of these differing spins would lead us to a position where the ball usually lands.

"Usually" could be "30% of the time it lands 10 pockets away from the fall point when the wheel is moving CCW at x/pi radians - or 8 pockets relative to initial ball velocity - and the ball's initial velocity is 3m/s..." Or any such analysis.

I could get much more specific but I agree that observations are necessary. I think Shannon and Thorp did manual clocking to begin with and accounted - with detail - for errors such as reaction time.

I would like to get information on the whole project from start to finish - pertinent details only - and would gladly pay a few bucks for papers or a book detailing this.

Can you enlighten me with what formulas one would use to calculate the deceleration of a ball traveling in a fixed circle with and without a friction co-efficient? And perhaps a formula for the vector function as the ball starts to fall after the deceleration comes to a point where the centripetal force becomes negative (? I could be using incorrect terms here!) or zero and falls down a slope - likely the slope has a slight curve - but it MAY be a plane?