Quote:MathExtremistWhat if I said that there's a roulette wheel in Nevada right now where the number 00 came up 9 times in the past 200 spins. Yes or no: is the wheel biased?

Maybe. But not likely.

Quote:MathExtremistI want to know what his tables say about the scenario where 00 comes up 9 times in 200 spins. Is that wheel biased or not?

From statmans tables it shows a 0.81660 probability.

So I do not think that would show a bias.

My sim shows 0.8358

But you already showed that his formula is about a particular number and not just any number.

ME, just give statman a quick PM and I am sure he would email to you his Excel worksheet and Kindle file of tables as he did for me.

I am in the middle of a large math and music project and just do not have the time, at this time, to pursue this matter.

I am always interested in unique math solutions to interesting math problems, that is the only reason I became involved.

Enjoy

Quote:guido111From statmans tables it shows a 0.81660 probability.

So I do not think that would show a bias.

My sim shows 0.8358

But you already showed that his formula is about a particular number and not just any number.

I'm interested in what is statman's interpretation of that number (whether it's 0.81 or 0.83) and what it means to his understanding of whether the wheel is biased.

Quote:guido111I am in the middle of a large math and music project and just do not have the time, at this time, to pursue this matter.

To completely hijack the thread, that sounds really cool. Are you doing something with algorithmic composition?

Quote:MathExtremistI'm interested in what is statman's interpretation of that number (whether it's 0.81 or 0.83) and what it means to his understanding of whether the wheel is biased.

To completely hijack the thread, that sounds really cool. Are you doing something with algorithmic composition?

Statman uses a 150 spin and 9 times hit as an example in his document that does not show a biased wheel but he really does not get into what would be a wheel bias as far as I understand.

He continues with "We would have to see 11 or 12 on the same number to call it unusual."

0.36028 = 9 hits

0.05660 = 11 hits

0.01806 = 12 hits

FYI: I do musical score arrangements and have done some interesting algorithmic compositions in the past but not currently.

It is fun stuff with the quality of today's digital keyboards.

Quote:guido111Statman uses a 150 spin and 9 times hit as an example in his document that does not show a biased wheel but he really does not get into what would be a wheel bias as far as I understand.

He continues with "We would have to see 11 or 12 on the same number to call it unusual."

0.36028 = 9 hits

0.05660 = 11 hits

0.01806 = 12 hits

So let's make it unusual: suppose you observe a roulette wheel for 150 spins and see the 00 show up 15 times. Now what? Does he conclude that the wheel is biased? Does he conclude that the likelihood of 00 appearing on the next spin is greater than 1/38?

If there is a wheel bias, for example because the slots are not all the same size, more likely than not, the error is caused by a misplaced single divider. As such, on that wheel, there would be a bias for a number, as well as a bias against the neighbor.

Additionally, if there is a wheel bias because its out of balance, then it will be biased for range(s) of numbers and/or biased against range(s) of numbers.

I doubt that his formulas account for either of these types of multiple biases.

Dealer ID | Wheel direction | Wheel RPM | p(theta between 0 and 45 degrees) | p(theta between 45 and 90 degrees) | ...

where theta is the angle between the spot on the wheel when the ball was released and where it lands. On a fair wheel, fairly dealt by a croupier, each probability will be 1/8 (give or take for the rounding issues). If you find one that's significantly greater than 1/8, there's a reasonable chance of a bias, and there's also a credible physical cause. It's not one that you'd ever detect by counting outcomes because theta isn't computed based on the actual outcome -- just the distance between the outcome and the release point.

Granted, you still need to be able to make bets after the ball drop, but you still can in most casinos. So, for example, if you have a dealer who tends to hit the octant between 135 and 180 the most when the wheel is spinning at a given speed, and you see the wheel spinning at that speed, wait for the ball to release and then put your money on those 5 numbers. Consider this a relative, table-lookup version of Eudaemonic Pie.

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This might be a bi-modal bias due to alternating use of large and small ball by the croupier.

>if there is a wheel bias because its out of balance,

This would be detectable by octet analysis.

Best bet: don't waste time trying to prove either of these very rare events.

Quote:statmanIn his paper, Murphy gives a circular chart showing unusually high frequencies for the the numbers 0, 25, and 29, however he doesn't say exactly what those frequencies were. That kind of information would be useful to me. Even with a perfectly true wheel it is possible to look back at the record and say you could have made money by betting on such-and-such a number, however with a perfectly true wheel there is no way of knowing in advance what that number would be. In order to make money consistently on a number, that number has to have a probability of coming up greater than 1/35.

MathExtemist says that your computations will be precise if you are working only with integers. That is true if they remain as integers, however the majority of mathematical environments have a limit on the size of an integer and if exceeds that maximum the variable will be converted to a floating point number and only the first fifteen or so digits will be retained. The rest will become zeros. The Python programming language has a large integer data type that will preserve all digits regardless of the size of the number. The computer algebra systems (CAS) also will do this. The top ones are Maple and Mathematica. These are pricey, but there are also some free ones. For details see the "Comparison of Computer Algebra Systems" topic in Wikipedia. The CAS's also will preserve the precision of rational and irrational numbers. If the square root of 2 is entered or is arrived at in the course of a calculation it will be kept as "sqrt(2)" in all further calculations. Sqrt(2) is accurate to an infinite number of decimal places. When the calculation is finished the CAS will approximate it to a number of decimal places specified by the user, which can be in the thousands. I recommend calculating an alternating series using a CAS lest some of the positive and negative terms cancel and produce garbage digits in the answer.

I don't know why I keep reading this junk.

Wait a sec...Quote:FleaStiff>If there is a wheel bias because the slots are not all the same size

This might be a bi-modal bias due to alternating use of large and small ball by the croupier.

>if there is a wheel bias because its out of balance,

This would be detectable by octet analysis.

Best bet: don't waste time trying to prove either of these very rare events.

Are you saying that a bias is a rare event, or that if there is a bias, it would be different than what I described?

If it's the latter, please elaborate, because the two types I described are all I can imagine.

Quote:statmanIn his paper, Murphy gives a circular chart showing unusually high frequencies for the the numbers 0, 25, and 29, however he doesn't say exactly what those frequencies were.

Yes he does. Re-read the article -- it's on the first page, bottom of the 2nd column.

Quote:MathExtemist says that your computations will be precise if you are working only with integers. That is true if they remain as integers, however the majority of mathematical environments have a limit on the size of an integer and if exceeds that maximum the variable will be converted to a floating point number and only the first fifteen or so digits will be retained. The rest will become zeros. [snip] I recommend calculating an alternating series using a CAS lest some of the positive and negative terms cancel and produce garbage digits in the answer.

You misquote me. I said "exact", not "precise". Exact has one meaning; precision is a relative measurement. Double-precision floating point representation is plenty precise to handle the sort of calculations we're talking about. The few results from you that I've seen, however, are incorrect by fully 1% or more. That cannot be explained by rounding error, but it is neatly explained by the fact that you used the wrong formula. See this post for the difference between "one specific number appearing N times in 38 sessions of M spins" vs "any of 38 numbers appearing N times in one session of M spins".

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sucker bought it and then complained that he only bought that piece of shit because he trusted this forum.

THIEF

Good name in man and woman, dear my lord,

Is the immediate jewel of their souls.

Who steals my purse steals trash; 'tis something, nothing;

'Twas mine, 'tis his, and has been slave to thousands;

But he that filches from me my good name

Robs me of that which not enriches him,

And makes me poor indeed.

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Here are three pieces of advice.

First, check your work via simulation. You have a fair wheel simulator, use it. You won't get close to 19.75%.

Second, re-read the Wizard's article on this problem. You've gotten things wrong.

Third, in view of the above, check your ego. Miscalculations and overwrought proclamations do not qualify you as one who can credibly lecture the members of this particular forum.

Quote:gogSo you are 'educating' people who make their living in part from calculating advanced mathematics/statistics, and come up with a method to beat an entire industry that makes their money off these numbers at their own game while using only a thousandth of the sample size, by copy and pasting sophomore year textbooks? And you are willing to share this knowledge with us for a bargain price of $25? I can't believe anyone can be this deluded, so that leaves only one other possibility.

THIEF, no more, no less !

The problem is "What is the probability of at least one number being unhit in 200 spins of a true roulette wheel?"

Answer: 0.1698457156512422824774366227571399727305239397937597577397915962140059107818329286640551237007763387

So, I'm guessing that what you are saying is that you are trying to detect a biased wheel through the occurence of non-hits based on a series of spins, and that theoretically, more non-hits means more hits, because if other numbers are hitting more, then other numbers are therefore hitting less.

Balderdash. Let's say that a wheel is biased towards three numbers, say 00, 0, and 1 with a frequency of .04 and that the frequency of all other numbers is .88 / 35 = .025143. Does the non-occurence of numbers now appear more frequently? Well of course it does, as the frequency of hitting the other numbers are less. But can you DETECT the non-frequency before detecting the bias. I say the answer is no.

In my little empircal analysis of 120,000 spins which represents 600 trials of 200 spins each where 00, 0, and 1 appear .04 of the time and the remainder appear evenly distributed:

No occurences of a number once occured on 98 occurences (16.333%). No occurences of a two numbers occurred 12 times (2%). No occurences of 3 numbers appeared twice (0.33%) . In the occurence where no occurences of 3 numbers, the most frequent numbers was not 00, 0, and 1, but 28 (10 times), 12 (appeared 11 times), 8, and 1 (appeared 10 times).

Inotherwords, to summarize, EVEN if you could conclude (and you cannot) that the non-appearance of numbers is correlated with a biased wheel, you cannot make that determination after 200 spins, and even if you could, the numbers that appear the most frequency in those 200 spins are not the ones that are biased.

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Quote:statmanNothing but rudeness and contempt! Very disappointing! Well, in case you think I can't calculate an accurate answer to Shack's problem, here it is to 100 digits. If you want 1,000 just say the word.

The problem is "What is the probability of at least one number being unhit in 200 spins of a true roulette wheel?"

Answer: 0.1698457156512422824774366227571399727305239397937597577397915962140059107818329286640551237007763387

So you mean "What is the probability of at least "ANY" one number being unhit in 200 spins of a true roulette wheel?"

Since there is a difference between a "specific" number and "any" one number when using a formula, your answer looks to be different from the answer in the table from your Kindle document.

Does this mean you now used a different formula?

The inclusion-exclusion formula that the Wizard used in his article appears to me to be the correct one to use for "any" number.

I also verified his results in Excel.

Your posted formula earlier in this thread that MathE explained looks to be for one "specific" number.

The results are very close but they are different and now I can see why others are bashing your math.

One must compare apples to apples and it looks like you are now doing that.

Yes?

Sally

For a complete discussion of the Wizard's problem look in his "Roulette" section. I am not especially interested in this problem but the hyenas here have been using his results to call me a hack. I now realize that the Wiz posts answers only to the problems he can solve. I posed my problem to him and instead of answering it he suggested I post it to the forum. No one on the forum knows how to solve it either. MathE makes a lot of noise but I have not yet seen him do a single calculation.

The question I posed to the Wiz was "What is the probability of at least one number coming up n times in b spins of a roulette wheel?" When he did not provide an answer I went to work on it myself. The results passed certain tests and since I had never seen this particular problem treated anywhere I thought I had something original. Now it appears that some of my figures may be off in the third or fourth decimal place. That doesn't matter for a statistical test that requires only two decimal places, but the hyenas are demanding perfection. My approach was to calculate the probability for a single number and then apply the union-of-events rule to include all the numbers on the wheel. This requires complete independence of the events and that may be the source of error.

The Wizard's result was easy to duplicate. There is a combinatorial function T(m,n) that gives the number of ways of distributing m distinguishable objects to n distinguishable boxes in such a way that no box is left empty. T(200,38)/38^200 is the probability that all slots will be filled. Subtract that from 1 and you have the probability that at least one will be left empty. I don't know how to apply this approach to the more general problem. I hope you didn't buy the table from Amazon. You can get it from me for free. I have also submitted it to www.freebookspot.es. I'll be glad to hear from you any time.

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Quote:MathExtremist: I'm interested in what is statman's interpretation of that number (whether it's 0.81 or 0.83) and what it means to his understanding of whether the wheel is biased.

If the tables say that the probability of a certain number of hits on a number in a given number of spins is 0.81, it means the probability of that happening by chance. For 0.81, that number of hits is very likely to occur by chance and does not indicate a biased wheel. A value of 0.00001, such as we get for Murphy's 0, means that that number of hits is unlikely to occur by chance and indicates a biased wheel.

Quote:statmanNo one on the forum knows how to solve it either.

Riiiiiight. Maybe they can't solve it. Or maybe they just think that you're a tool and the problem isn't worth their time. Seeing as how at least 3 members of this forum are widely respected professional gaming mathematicians, I'm guessing that the latter is the case...

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Quote:statmanrdw4pous: "at least 3 members of this forum are widely respected professional gaming mathematicians"

Would you please identify them? I'd like to know more about them and read their posts. Give a brief bio if you would!

I was thinking of the Wiz, Math Extremist, and Teliot. You can do your own research on their bios. Based on previous conversations, I think Cindy Liu might also be a member, but I don't know what her handle is.

Quote:rdw4potusRiiiiiight. Maybe they can't solve it. Or maybe they just think that you're a tool and the problem isn't worth their time. Seeing as how at least 3 members of this forum are widely respected professional gaming mathematicians, I'm guessing that the latter is the case...

Or maybe the solution has already been posted, and maybe at least two of those professional gaming mathematicians are wondering why this conversation is still going on...

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Quote:statmanrdw4pous:MathExtremist has contributed extensively to this thread and he has shown me about as much knowledge of math as the average high-school dropout, however a non-mathematician might be fooled by his hubris.

Wow.

Maybe you have not read all of his [MathExtremis] posts here at WoV.

I now see, after reading all the posts in this thread since so many for yours are missing, why so many here are against you.

You started off saying something about a fantastic discovery and then the math behind that discovery was found to be actually answering a different, but close, question. A "specific" vs. "any" number.

The more I have looked into this, yes this makes for some interesting math. I am not that good at math so no help can come from me.

MathExtremist gave his solution for your question as a multinomial distribution, etc, he lost me on the rest.

I just do not think he wants to spend the time in solving your question as you may run off with the solution and claim it as your own. I am not saying you would do that, but this is the 21st century and the internet has shown that this does actually take place.

Maybe you should have just asked for help on a great math question without the biased, non-biased wheel being thrown into the mix.

My BF says your math question might actually be easier to solve using a Markov chain.

I can do real basic ones but not for your question. Maybe CrystalMath or weaselman, just to name a few, would offer their help. They have shown their excellence in using math skills.

But then, maybe they do not like you by how you have behaved here.

just my 2 cents

Sally

Quote:DJTeddyBear

Admin note: removed image www.djteddybear.com/images/wheel_analysis.JPG

The piece of the puzzle that I can't read is the number of spins in this analysis....

I can read the following from your image:

Data Series: Last 20,000 spins

Start Date: ??/??/1999

End Date: ??/??/2000

Wheel Type Single Zero

Num Spins 20,004

Mean 541(?)

%Std Dev 2.5%(?)

Std C?? Test CAUTION

Max Z Score 2.34(?)

Num Test FAIL

While this is clearly useful data for the casino, it is presumably collected electronically by those notoriously inaccurate scanners over the wheel.

I would think it is also likely the data collection has errors - sometimes the same number appears repeatedly after just 1 spin.

However the shape of the graph in this case does indicate bias towards a part of the wheel.

Even so, what player is going to record 20,000 spins to find this out at the same time the casino is finding out and fixing it?

Quote:mustangsally

MathExtremist gave his solution for your question as a multinomial distribution, etc, he lost me on the rest.

I just do not think he wants to spend the time in solving your question as you may run off with the solution and claim it as your own. I am not saying you would do that, but this is the 21st century and the internet has shown that this does actually take place.

No, I don't want to spend the time because I'm at G2E meeting with clients and licensees. I'm a consultant - I only take abuse from people who pay me. :)

Quote:MathExtremistI only take abuse from people who pay me. :)

And here I thought that was illegal in Clark county...;-)

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Quote:statmanIf we are evaluating outcomes at the 5% level of significance, 1% accuracy in the tables is more than enough.

Thank you. I needed a laugh this morning.

This question is to be found on https://wizardofodds.com/ask-the-wizard/roulette/

What are the chances of a dealer hitting 5 of the same number in 10 spins of the roulette wheel? -- Spence from Red Deer, Alberta

Wiz: The chances of any number occurring exactly 5 times in 10 spins in a double-zero roulette game could be closely approximated by 38*combin(10,5)*(1/38)5*(37/38)5 = 1 in 359,275.

Apparently he has a way of figuring these things out, but we are far apart. His answer is 0.000002783, My answer is 0.00011.

We also are apart by a factor of 100 on a subsequent question: 6 hits in 1000 spins. My tables say the probability is 0.00005

Quote:statmanWho is Right? (Or who is closer to the correct value?)

This question is to be found on https://wizardofodds.com/ask-the-wizard/roulette/

What are the chances of a dealer hitting 5 of the same number in 10 spins of the roulette wheel? -- Spence from Red Deer, Alberta

Wiz: The chances of any number occurring exactly 5 times in 10 spins in a double-zero roulette game could be closely approximated by 38*combin(10,5)*(1/38)5*(37/38)5 = 1 in 359,275.

Apparently he has a way of figuring these things out, but we are far apart. His answer is 0.000002783, My answer is 0.00011.

We also are apart by a factor of 100 on a subsequent question: 6 hits in 1000 spins. My tables say the probability is 0.00005

I do not know who is closer to the exact answer. Where is the exact answer?

The Wizards formula and his 1 in do not match. No one is perfect all the time. He also states it is approximated.

38*combin(10,5)*(1/38)^5*(37/38)^5 = 0.000105769 or 1 in 9454.6 (Maybe he was changing a diaper at the time and was distracted by a phone call)

Your table answer is 1 in 9090.9

My 10 million sim is 0.0001087 or 1 in 9199.6

I say we all win a prize!

There is no doubt that the exact math formulas for these type of questions is very challenging.

Since you use Excel and know how to solve this with inclusion-exclusion, a process very time consuming and like you said before, possibility of errors,

why not just set up a macro in Excel to run the calculations? That way you can correct any minor mistakes you may find and you will finally have the exact answers you are looking for instead of "just close" answers that may not always be that close.

You will be the first.

I say go for it!

added a few minutes later:

From Statman's previous previous post:

"Would the above be worth the extra time and trouble to program

and execute seeing that the difference between the approximate and exact values appear to be

only about 1% "

I say yes! Unless you know that ALL your results are that close, better to be perfect if given the choice.

I work with DNA researchers, no I am not one of them, and they freak out when something in their data is off by .001%. Of course they deal with sample sizes of a few billion so I do understand them wanting to be exact in their calculations.

Quote:statmanWho is Right? (Or who is closer to the correct value?)

This question is to be found on https://wizardofodds.com/ask-the-wizard/roulette/

We also are apart by a factor of 100 on a subsequent question: 6 hits in 1000 spins. My tables say the probability is 0.00005

I missed the last question from the WoO site.

"I played the same number 1000 spins in a row on a 0,00 wheel and hit 6 times. What are the chances of hitting 6 or fewer times in this scenario? Thanks, Bill K."

This is just a binomial distribution question. And Bill is looking for a "specific" number not any number. Right?

In Excel =BINOMDIST(6,1000,1/38,TRUE)

1.77565E-06 or 1 in 563175.4617

The Wizard continues "So the answer is 0.00000177564555, or 1 in 563175. I hope this didn't happen at an Internet casino"

So, when you ask, what are the odds of number x hitting n times in b spins, it's 1/38 of the odds of any number hitting n times in b spins.

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Quote:statmanI hesitate to find fault with the Wizard's work. It would be like criticizing God.

If you can find a problem with his work, he is very humble and appreciative. If he wanted to back you up, he would be on here.

As for you insulting people (such as MathExtremist), that is very distasteful.

Suppose a multinomial experiment consists of n trials, and each trial can result in any of k possible outcomes: E1, E2, . . . , Ek. Suppose, further, that each possible outcome can occur with probabilities p1, p2, . . . , pk. Then, the probability (P) that E1 occurs n1 times, E2 occurs n2 times, . . . , and Ek occurs nk times is

P = [ n! / ( n1! * n2! * ... nk! ) ] * ( p1n1 * p2n2 * . . . * pknk )

where n = n1 + n2 + . . . + nk.

You ask, what is the probability that a result is on any number 5 times in 10 spins, and the Wiz used the binomial distribution to calculate the result for the number 1 and then multiplied it by 38. It's an approximation because it is possible for say example 00 to occur 5 times and 0 to occur 5 times. He multiplies the result by 38 to get the approximation.

The correct analysis would use the mathemically sound multinomial distribution, but the number of iterations required on a roulette wheel to get all possible answers would be a very long one indeed, not impossible, but definitely programmable, and long.

The point that mathextremist, crystalmath, and I are making is that there is nothing new here.

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Quote:statmanFor those who have access to Jstor, the link is http://www.jstor.org/pss/2287733

If I can get the other pages, I'll post them as well. If anyone is interested in Ethier's book, let me know. Discount what Scarne says. He didn't believe in Thorp's card counting strategy.

S.N. Ethier's Book

The Doctrine of Chances 2010

chapter 13.2

"Suppose we have “clocked” a roulette wheel for n coups, and the most

frequent number has occurred a proportion 1/25 of the time. Is it a biased

wheel?"

Excellent reading and he even appears to answer the question

He is a top notch Math Professor

added: in the same book

"13.16.

Number of coups needed to conclude that a favorable number exists.

We saw that, if the most frequent number in n coups has occurred

a proportion 1/25 of the time, we can conclude with 95% confidence that

one or more favorable numbers exist if n ≥ 1,618."

Okay. Much of this thread consists of five zero posts which appears to reflect some Wiki Memory Wipe of someone who wanted to take his toys and go home.Quote:statman00000

The rest of the posts seem to be something about multinomial distributions are arguments of numbers hitting and numbers not hitting and what can considered proof of something versus evidence consistent with it, but not proving it.

I'm lost on much of the math anyway.

May I propose a practical trial?

Will someone who knows what he is talking about and does not believe in the utility of these much touted tables simply get a set of the tables and sit down at one of those cheapie roulette wheels that require a grand total of fifty cents on inside bets or something. Clock the wheel for an hour, then play according to the tables for an hour... and let me know whether you win or lose.