Lottery Probability

[Jay-qu]

well you could say that the coin has a 4th option - it may quantum tunnel through the earth - and a 5th option... and so on, you can go on to infinity with that point :confused:

[C1ay]

Chaos says the game isn't fair because you can predict short term by analyzing the past performance of a system & looking for islands of stability - attractors if you will - that have relatively predictable short term patterns.

It is important to note however, many lotteries use a new set of balls for each drawing to prevent any of the balls' properties from contributing to any such islands of stability.

[Turtle]

well you could say that the coin has a 4th option - it may quantum tunnel through the earth - and a 5th option... and so on, you can go on to infinity with that point :confused:

Yes exactly; that is just the point. Those possibilities really exist. Probability came out of trying to understand the occurence of real events (OK gambling) & despite theoretical work, that is entirely what it is about.

Here is a poser; in the 6/49 Lotto game, is a 6 number combination of all even numbers equally as likely as one of all odd numbers?

I have to refer to this article again as they make some of my points in a different manner:

http://www.cs.brown.edu/research/ai/dynamics/tutorial/Documents/CrackingWallStreet.html

:confused:

[CraigD]

Keep the 1 2 3 4 5 6 & choose 5 10 15 20 25 30 for comparison, or more precisely any 6 number combination where all elements are multiples of 5

…

_______For 1 2 3 4 5 6: Probability is 6/49*5/48*4/47*3/46*2/45*1/44

___For 6 multiples of 5: Probability is 9/49*8/48* 7/47*6/46*5/45*4/44

 

____Clearly these 2 combinations don't have the same probability.QED

Correct. However…

 

In general, there are m!/(n!*(m-n)!) ways of choosing n of m elements where order isn’t important (where “!” is the factoral unary operators, eg: 4! = 1*2*3*4) [1].

 

The probability P of drawing a specific n of m elements is its reciprocal, (n!*(m-n)!)/m! [2].

 

The probability of choosing any 1 of x different choices of n of m elements is x*P [3].

 

As you’ve correctly calculated, and per [1], the probability P1 of drawing exactly {1,2,3,4,5,6} is (6!*(49-6)!)/49!,

while the probability P2 of drawing any 6 balls numbered with multiples of 5 is (9!*(49-6)!)/(49!*(9-6)!).

 

Note that P2/P1 = ((9!*(49-6)!)/(49!*(9-6)!)) / ((6!*(49-6)!)/49!)

= (9!*(49-6)!*49!)/(49!*(9-6)!* (6!*(49-6)!))

= 9!/((9-6)!*(6!))

= 84

 

However the probability of choosing a specific 6 of 49 numbered balls (eg:{5,10,15,20,25,30}) is not the same as choosing any 6 of 49 where each’s number is a multiple of 5 (eg: {5,10,15,20,25,30} or {5,10,15,20,25,35} or {20,25,30,35,40,45}, etc.)

 

There are 9!/ ((9-6)!*6!) = 84 possible ways to draw 6 of 9 (the number of ) elements. As expected from [3], this is the same as P2/P1.

 

So, in a fair drawing of 6 of 49 balls, the probability of drawing {1,2,3,4,5,6} is the same as the probability of drawing {5,10,15,20,25,30}, or any other choice of numbers – 1/13983816.

[Turtle]

___I still maintain that is misleading in regard to modeling & predictability. I mean to provoke a response on the significance of modeling any given system - from Brownian motion to insurance rate tables - by equations from chaos theory and not from probability theory. If one is better, i.e. more accurate in prediction - than the other, then using the lesser is illogical.

___I think the evidence is clear, if relatively unknown, that chaos is the better. I only wish to corrupt the youth. :confused:

[cwes99_03]

Here is a poser; in the 6/49 Lotto game, is a 6 number combination of all even numbers equally as likely as one of all odd numbers?

 

No, there are more odd numbers in total. 25 odd, 24 even

24/49,23/48,22/47,21/46,20/45,19/44 = (24!-18!)/(49!-43!)

25/49 24/48 23/47 22/46 21/45 20/44 = (25!-19!)/(49!-43!)

Meaning odds are hehe that the list of odd numbers is more likely to get you to win than the list of even numbers.

[Turtle]

No, there are more odd numbers in total. 25 odd, 24 even

24/49,23/48,22/47,21/46,20/45,19/44 = (24!-18!)/(49!-43!)

25/49 24/48 23/47 22/46 21/45 20/44 = (25!-19!)/(49!-43!)

Meaning odds are hehe that the list of odd numbers is more likely to get you to win than the list of even numbers.

___Now that...is brotherly conduct. :confused:

[Erasmus00]

___I still maintain that is misleading in regard to modeling & predictability. I mean to provoke a response on the significance of modeling any given system - from Brownian motion to insurance rate tables - by equations from chaos theory and not from probability theory. If one is better, i.e. more accurate in prediction - than the other, then using the lesser is illogical.

___I think the evidence is clear, if relatively unknown, that chaos is the better. I only wish to corrupt the youth. :confused:

 

It depends on the system. In many instances, standard random walk works wonderfully for brownian motion. In the lottery as well, standard probability theory works pretty well (mostly because they do everything possible, from swapping out balls and machines to make the game as fair as they can). Probability theory also has the tremendous advantage of allowing you to calculate things from first principles.

 

Now, use the same roulette wheel and ball over and over again, and chaos theory is your game. Or with the stock market, where you have loads of past data to stick into models...

-Will

[Turtle]

Now, use the same roulette wheel and ball over and over again, and chaos theory is your game. Or with the stock market, where you have loads of past data to stick into models...

-Will

Or insurance data, medical data, psychological data, geological data, etc.. In short, everything we've been using standard probability to model - inadequately model. These areas affect real peoples lives in their economic situation, their health, their rights accorded by government & law, & other numerous aspects. I feel that where this new knowledge is not applied, whether by ignorance or deceit, we the people get screwed. Notice the air of secrecy in the article I linked to? Notice this guy was at Los Alamos? Notice his current contract won't allow him to say just how accurate the predictions get? Notice these chaos algorthms have world consequrnces? Follow the money. Not only who is spending it, but on what.

I have more, but I need a fresh bier. Discuss. :confused:

[Erasmus00]

Or insurance data, medical data, psychological data, geological data, etc.. In short, everything we've been using standard probability to model - inadequately model. These areas affect real peoples lives in their economic situation, their health, their rights accorded by government & law, & other numerous aspects. I feel that where this new knowledge is not applied, whether by ignorance or deceit, we the people get screwed. Notice the air of secrecy in the article I linked to? Notice this guy was at Los Alamos? Notice his current contract won't allow him to say just how accurate the predictions get? Notice these chaos algorthms have world consequrnces? Follow the money. Not only who is spending it, but on what.

I have more, but I need a fresh bier. Discuss. :confused:

 

Lots of physicists (even those who haven't left the field) have been applying chaos theory to all the things above. Well, perhaps not psychological or medical data (I fail to understand the connection) but certainly to geological data, insurance data, etc. Scientists and private industry alike are trying to predict the stock market, lightning, turbulent fluid flow, earthquakes, etc.

 

As to the secrecy, the guy in your article wasn't at a government facility, he worked for a private research company, and his contracts were with private companies who didn't want their golden goose plucked. If everyone suddenly used his methods to predict the stock market, the market would adjust, and the system would change, rendering his predictions useless.

-Will

[Turtle]

Well, perhaps not psychological or medical data (I fail to understand the connection) but certainly to geological data, insurance data, etc. Scientists and private industry alike are trying to predict the stock market, lightning, turbulent fluid flow, earthquakes, etc. -Will

 

Why should medical or psychological data get different treatment? When good old Doc says oh, 30% chance to die, or recover, or whatever, isn't that important enough to get the same analysis as my car insurance?

 

___As to the guys system going useless, that only means you have to put the changed system data back in. You make it sound like it is an artifice somehow Erasmus. Maybe the geologists work is too arcane for me to have read that uses chaos; I don't readily recall reading any.

___Perhaps it is not a problem that it isn't widely used, but rather not widely published. Any links at hand? :confused:

 

Addendum: I only had to look. :confused:

http://www.cambridge.org/uk/catalogue/catalogue.asp?isbn=0521567335

[CraigD]

___I still maintain that is misleading in regard to modeling & predictability. I mean to provoke a response on the significance of modeling any given system - from Brownian motion to insurance rate tables - by equations from chaos theory and not from probability theory. If one is better, i.e. more accurate in prediction - than the other, then using the lesser is illogical.

___I think the evidence is clear, if relatively unknown, that chaos is the better. I only wish to corrupt the youth. :confused:

My take on chaos theory is that it’s not better than probability theory, but supportive of probability theory’s fundimental postulates.

 

To apply to the physical world, probability theory depends on the existence of randomness, which, in a non-chaotic system, doesn’t exist. For example …

 

Imagine a typical gravity-drop 6/49 Lotto machine. To simplify this example, imagine that, rather than being operated by a human being, it is attached to a mechanical timer, let’s say a large, open grandfather clock with the simplest practical pendulum, escapement, and gears. At the scheduled time of the Lotto drawing, this combined machine releases the balls, stirs them, collects the 6 winners, then ceases to be relevant to this example.

 

If this system – Lotto machine, balls, timer, intervening air, light, gravity, earth tremors, etc. – were not chaotic, a person with conceivably precise measuring equipment could, from a measurements made of the balls, Lotto machine, timer, and all the other significant parts of the system, calculate before the drawing the precise trajectories of every ball, and the resulting 6 winners. Even if his equipment was not sufficiently precise to predict the 6 winners with certainty, he could assign different probabilities to each ball following a trajectory that would land it in the winner’s chute, allowing him to buy a collection of tickets that would give him an aprobabilistic chance of winning.

 

Chaos theory says this is not the case. Specifically, it says that improvements in the accuracy of the cheater’s measuring instruments won’t improve the predictive accuracy of his calculations. In more chaos theoretical language, it says the size of the error in measurement of the system’s initial conditions is not a predictor of the accuracy of the size in the error of his prediction vs. the actual outcome. In short, chaos theory says that randomness effectively exists, even in a purely deterministic mechanical system.

 

So, rather than providing a way to make Lotto picks with better-than-probabilistic chances of winning, chaos theory says it’s futile to try.

 

Chaos theory does go further, to describe ways to calculate around this “critical sensitivity to initial conditions”, providing alternatives to classical mechanical models. These methods involving things like “attractor basins”, but chaos theory by no means guarantees that every chaotic system has an attractor, or that the ones it has are useful for a particular predictive purpose. Other than making the obvious prediction that, in a gravity-prop Lotto machine, some balls will eventually make it to the chute – a prediction that would be difficult to make using pure mechanics – chaos’s theory of attractors doesn’t appear to provide any way to predict which 6 balls will reach the chute first.

[cwes99_03]

The most likely scenario for any 6 numbers is that an equal number of them will be even and odd, or 4 will be odd and 2 even.

 

25/49*24/48*24/47*23/46*23/45*22/44

 

Notice that after the first ball being an odd, the next could be an odd or even, the one thereafter would be most likely the opposite of whatever the second was and so on and so forth. then the 6th ball picked has equal chances of being odd or even. If it is even then there are 3 odd and 3even. If it is odd there are 4 odd 2 even. These two possibilites have the same probability.

[Turtle]

So, rather than providing a way to make Lotto picks with better-than-probabilistic chances of winning, chaos theory says it’s futile to try.

___Whaaa!? That doesn't seem to jibe with what the article says about Farmer's work:

 

Farmer's computer algorithm was based not on the mathematics of roulette but on the physics of the wheel. In essence, the cabal's code simulated the entire rotating roulette wheel and bouncing ball inside the chip in the shoe. And it did this in a minuscule 4 Kbytes of memory...

 

...When everything worked, the chips won. The system never predicted the exact winning number: the cabal members were realists. Their prediction machinery forecast a small neighborhood of numbers -- one octave section of the wheel -- as the bettable destination of the ball.

 

____Because these guys beat the roulette wheel, & because a roulette wheel is considered 'random', i.e. not predictable, then I'd say chaos has something over probability here. The same for the Lottery, bouncing balls & all. The main difference I see is that real data - & the most current real data- from the system - is required to apply the chaos techniques. You have to know what's gone on before in the system & what's going on right now, & islands of opportunity appear & disapear as the playing progresses. You have to watch, recognize an opportunity, & sieze it before it passes, & you have a better chance to win than otherwise. :confused:

[cwes99_03]

Thus is the basis for one of the best shows on network TV at this time, Numbers Friday nights on CBS. I love that show.

[CraigD]

So, rather than providing a way to make Lotto picks with better-than-probabilistic chances of winning, chaos theory says it’s futile to try.

___Whaaa!? That doesn't seem to jibe with what the article says about Farmer's work…

There are significant differences between predicting the approximate ending location of a roulette ball after observing the wheel’s spin and the balls launch (what Farmer, Packard, and his “nerd hippy” companions did so famously in the 1977) and predicting the order in which the balls in a Lotto machine will fall into its chute a day or more in advance.

 

Farmers scheme is much like predicting the resting place of a baseball thrown in an empty, not smooth but not-too-lumpy field. By knowing something of the physics of the bouncing ball, and making a reasonably measurement of the position from which and speed and angle with which it thrown, one can predict of where it will come to rest within +- some reasonable error margin. This is roughly analogous (though easier) to what Farmer’s team did with their purchased 2nd-hand roulette wheel. It’s impressive that they were able to judge the initial wheel and ball speeds and position well enough for the scheme to work – I’d suspect that many human perception distractions could confuse an the observer from doing this – though, really, the data for how well it actually worked is very anecdotal – a few paragraphs in books like Kelly’s ”Out of Control” (whole book available via link!) or Gleik’s excellent ”Chaos:Making a New Science” (which I know of only in paper).

 

A major point Farmer makes in even these brief passages is that, to circumvent the randomizing effects of chaos, it’s futile to try to look too far ahead. In the roulette scheme, they considered only the immediate data for one spin/launch of the wheel/ball in their (computer concealed in a shoe!) calculations. Past history of the wheel – its previous numbers - are not considered at all. This is markedly different than the approach most “win the lottery” analysis schemes approach prediction Lotto machine outcomes.

 

A similar scheme with a Lotto machine would be to observe the machine in action with a high-speed camera, detecting the numbers and motion of the balls before they enter the chute. I’m confident that, using existing video, pattern-recognition, and modeling software, such a system could be made, and would work, allowing one to know Lotto numbers perhaps even seconds before everyone else. How much further in advance would depend on the goodness and sophistication of your (almost certainly chaos theory using) modeling software. Unlike roulette, where bets can be placed seconds before the ball lands, you have to make Lotto number picks well in advance of the drawing, so the scheme wouldn’t be of much practical (eg: get rich) use.

[Erasmus00]

Why should medical or psychological data get different treatment? When good old Doc says oh, 30% chance to die, or recover, or whatever, isn't that important enough to get the same analysis as my car insurance?

 

Its a different problem though. There is 0 data as to how YOUR body will behave under some particular illness or trauma. (because you've never been diagnosed with that problem or trauma) There is loads of data, though, on how lots of different people have responded to the condition. So here, the postulates of probability are the correct ones.

-Will