Roulette player vs Math professor (a dialog)

Gambling expert R.D. Ellison vs Math Professor J. Laurie Snell (1)

These are undoubtedly three of the best discussions ever published about the theory of roulette and the meaning and philosophical aspect of probabilities. The participants are intelligent, knowledgeable and civilized, yet they approach probabilities from completely different angles which makes for an extremely interesting discussion. This is an intellectual tour de force that reminds me of Socrates’ dialogues. Enjoy!

R.D. Ellison

Dr. J. Laurie Snell

Note: To help clarify who is talking, Dr. Snell’s words are shown in yellow.
The messages were edited for brevity; I removed the “hello”, “thank you”, “regards” etc.

R.D. Ellison
Dr. Snell,
I am the author of three nonfiction books that delve into the subject of probabilities. My research in that area has led me to discover some serious contradictions in one of the most widely accepted probability theories of our time. One of the mathematicians I contacted to discuss this, Joan Garfield, suggested that you might be able to help me.
I am seeking someone who understands probabilities to evaluate my findings. I would sincerely appreciate any counsel you could offer.

J. Laurie Snell
I would be happy to look at your work.

R.D. Ellison
Hello Dr. Snell,
Thank you so very much. You have truly lifted my spirits!
I think the best way for us to begin is for you to review a 685-word article I have posted Gambler’s Fallacy Debate
What I’m seeking is an appraisal of my position on the “independent events” issue.

R.D. Ellison
I realize that my request to which you replied ten days ago involves a somewhat complex issue. So I want you to take the time you need to fully look into it, but at the same time, I have some measure of insecurity as to whether I will hear from you again. This is derived from past experience, which has taught me that most people prefer not to challenge the established premise, and they don’t want to debate the issue, either. They just want to retreat to the “safety” of the existing logic, and turn to other matters.
So, I am asking for an interim reply that will let me know if you are still looking at this. If not, I would appreciate an opportunity to present my case, if you remain unconvinced by what you’ve seen so far.

Sorry. When I am trying to write my newsletter I tend to forget everything else.A mathematical model is just a model and no guarantee that it will apply to the real world. For example, the standard model for the toss of a coin is independent trials with probability 1/2 for heads on each toss. Of course we know that coins are not perfectly made and the model may not apply to every situation. For example if you stand pennies on their edge on a table and bang the table heads will come up about 70 to 80 percent of the time. However, if you really toss it thousands of times the outcomes do seem to fit the model pretty well.

The best way to test if a model applies is to do experiments. For example the standard model for coin tossings predicts that when you toss a coin 100 times there is a 95% chance that the number of heads will not differ more than 5 from 50. To check this we toss a coin 100 times many times and see if only about 5% of the time is the difference more than 5. As more of our theoretical results are verified by experience we have more confidence in the model. Experience with this when you actually tossed a coin do suggest that the standard coin tossing model is o.k.

If you want to show that the standard model for roulette which proves independence is not correct you should propose a specific experiment that you feel would not come out as expected by the independence model. Then we can try it out and see what happens. For example, if you believe that the roulette wheel has been constructed in a way that would lead it to equalize the number of black and red outcomes differently from that predicted by the standard model we can try this out and see if there is a difference.

When we do clinical trials we certainly hope that the pills are different from placebo that we model by coin tossing. However, if experiments show that when equal numbers of people are given the pill and placebo the results for the pills cannot be distinguished from the coin tossing model we tend to reject the idea that the pills are helpful. Without experiments we would not be able to make any evaluation of the drug.
I am sure you have heard all this before but this is the best I can do.

R.D. Ellison
The problem I face is that the accepted logic is so well entrenched, people accept it without realizing it contradicts itself. An example of this sort of thing is the current flap over saying the Pledge of Allegiance in schools. A Federal court in California has ruled it unconstitutional, saying that it violates the separation of church and state. There are many different gods, they have said. Why not “under Buddha” or “under Zeus”? Technically they’re right, but saying the words “under God” in that context, isn’t likely to whip someone into a religious frenzy, so what’s the harm, the detractors say. Point is, it’s been accepted for so long, nobody gives it a thought.
And so it is with the “independent events” issue. Nobody has stopped to realize that assigning a statistical expectation to a series of independent events is a humongous contradiction. You can’t “expect” any behavioral pattern from something that is independent. The word “independent” means “free from influence.” Anything with an expectation is being influenced! As such, any two entities that are mutually exclusive cannot be merged into a single concept.
You talked about the difference between small samplings and large samplings: “if you toss it thousands of times the outcomes seem to fit the model pretty well.” Thus, you expect the numbers to conform in large groups but not in small groups. This is also a contradiction. An accumulation of small groups will form a large group, therefore the statistical pressure for numbers to conform to their probabilities is actually felt in each and every one of those thousands or millions of numbers that form the large group. For lack of a better expression: each number is a tiny, tiny part of a greater conspiracy that will ultimately reveal itself as the trials accumulate. What this means is that these numbers are conforming, every step of the way. And something that conforms is in defiance of the core meaning of independence!
I could go on with more examples, and I can also give you some experiments to try, but I don’t want to burden you with too much information at once. But I hope and trust that you can see that I have some good points.

J. Laurie Snell
I agree that you have an interesting philosophical point about the nature of probability. The fact that the law of large numbers allows you to predict very closely the proportion of heads that will come up in the long run does not seem consistent with the intuitive notion of independence. However, this does not say that the standard mathematical model for probability is wrong. It is just a mathematical theory and as I have said before, we can only decide if it fits a particular physical process by carrying out experiments.

So again it would help me understand your concern if you would give me predictions regarding the outcome of roulette or some other gambling game that is inconsistent with the predictions of probability theory. Then we can carry out the experiment and see if you are right.
Of course, we can also argue about philosophy but that is not something I enjoy and in my declining years I try to avoid things I do not enjoy.

R.D. Ellison
I understand your reluctance to chart a course toward a philosophical quagmire. And I realize that many people before me have challenged this issue, and all they accomplished was to usurp the time of those who were kind enough to listen. With that in mind, I assure you that I will stick to the facts, and get quickly to the point in each case.

I appreciate your preference of utilizing a model to provide a factual resolution to the issue. But 10 billion trials would probably be insufficient to prove my contention that no roulette table could produce 60 consecutive wins by an even-chance wagering proposition. Because, although no one has ever seen this happen (there is no record where even half that figure was ever reached), math theorists will maintain that it is possible. This sort of debate would lead us straight towards the philosophical dead end we are trying to avoid.
I believe we can accomplish much more if we establish a question-and-answer format that emphasizes brevity. This will keep it simple, and enable us to move quickly from point to point. This is also designed to ensure that we concur on the meaning of the terms we use. For the moment, all I am seeking is a one-word answer to each of the three questions below. Please feel free to elaborate if you think a clarification would be helpful:
1) As I understand it, you concur with the premise that if a small sampling of unbiased even chance trials starts out off-balance (e.g., nine out of ten coin flips are tails), that as the number of trials increases, you will see a decisive inclination toward a state of equality in the number of “wins” for the two sides? Yes or No
2) In accordance with the prevailing wisdom, do you also concur with the premise that the roulette wheel, the dice, etc., have no memory, or capacity for cognitive thought? Yes or No
3) If you agree with question 2, does it then logically follow that any numbers generated from those devices, assuming there is no bias, should be regarded as independent events? Yes or No
Please bear with me. There is a grand design to this line of questioning, which will become evident in due course. Thank you very much for your time. Sincerely,

J. Laurie Snell
— You wrote:
1) As I understand it, you concur with the premise that if a small sampling of unbiased even chance trials starts out off-balance (e.g., nine out of ten coin flips are tails), that as the number of trials increases, you will see a decisive inclination toward a state of equality in the number of “wins” for the two sides? Yes or No
NO
2) In accordance with the prevailing wisdom, do you also concur with the premise that the roulette wheel, the dice, etc., have no memory, or capacity for cognitive thought? Yes or No
YES
3) If you agree with question 2, does it then logically follow that any numbers generated from those devices, assuming there is no bias, should be regarded as independent events? Yes or No
NO
Please bear with me. There is a grand design to this line of questioning, which will become evident in due course.

R.D. Ellison
Thank you so much for your timely reply. I need a clarification, however, because what I see below is not what I wrote. Specifically, the last line of question number 1 contains the word Nwinsa. In my email, the word was wins, surrounded by quotations. (I am using bold text instead of quotations this time, for fear of it happening again.)
I have seen this problem before. Some ISPs go haywire when translating punctuation from the clientele of other ISPs, particularly apostrophes and quotations. Be that as it may, I need to confirm your answer for question number 1, so that I have assurance that you are responding to the question that was submitted. Thank you.
J. Laurie Snell
Yes my answers were
(1) No
(2) Yes
(3) No

R.D. Ellison
I have to admit, you really threw me with your answer to Question number 1. To make sure we are talking about the same thing, I would like to ask for a confirmation of something slightly more specific. Let us first establish that all questions assume that the devices used to obtain these statistical results are not biased from any mechanical defects or external influence. And, in the following case, there are only two possible outcomes: a decision of heads, or tails. As for the question itself, let us go with the premise that in a given sampling, nine out of ten coin flips came up tails. I will contend that from that point forward, given enough trials (an important point), you will see a reduction in the 90 percent ratio that favors tails. And that this will happen every time, without exception. Do you disagree with that supposition? Note: a Yes answer will mean that your reply to Question number 1 remains unchanged, and that your reply to the more specific version, above, is also No. If you have changed your mind, or offer different answers to the two questions, please clarify.

Regarding Question number 2, that is the answer I expected. We will come back to that.

Regarding Question number 3, once again, you have offered an answer that I did not expect, because it conflicts with my understanding of what most experts believe, that one of the primary reasons that table results at roulette are considered to be independent events is because the wheel, having no memory, is incapable of remembering what happened in the past, and therefore, those numbers cannot be subjected to any influence, such as having a reason to be due. Again, I am seeking a clarification: having reviewed these comments, is your answer to Question number 3 still No? And, if that is the case, you must believe that there is some other reason that table results are considered independent events, which eclipses the aforementioned reason. Could you impart to me that viewpoint?

R.D. Ellison
I think we are making progress! We both concur on Question 1 (reconfigured), and Question 2. The former states that as long as there is no bias to deal with, a coin toss sampling will never favor one side by a ratio of 90 to 10 on an ongoing basis. Question 2 states that precision-made instruments like a roulette wheel or pair of dice have no capacity for cognitive thought, and therefore are bereft of ‘real’ memory.

I have tried to phrase the above paragraph so that the intent of the words is unmistakable, but please feel free to recommend changes if you feel it necessary or prudent.
Regarding Question 3, we have a slight snag. Three, actually. The first is that your reply is not unmistakably clear to me. I hope you will agree that there needs to be absolute clarity on both sides. The second snag is that the issue is connected to a question to which you have not responded. That is, if it isn’t the memoryless aspect of the roulette wheel that makes these events “independent,” then what does? Now, if you don’t believe they are necessarily independent, then we have a whole new ball game. In either case, I would appreciate a clarification.

This third snag pertaining to Question 3 is your mention, once again, that you prefer models or experiments over dialogue as a means of establishing proof. As well-intentioned as this approach may be, I think it may add an element of impossibility to my task, because you are seeking a finite answer in a situation that does not yield finite results. For example, there is no record of any roulette table ever producing 25 of the same even chance results in a row, but most theorists, I believe, would contend that it is possible. So, at what point do we concede that enough trials have been performed to constitute proof? 5 billion may not be enough, but one trillion might actually produce such a table. The trouble is, neither you or I will live long enough to resolve it!

So, I am asking you to please make an effort to work with me here, and to not impose restrictions that guarantee failure on both our parts. What I am trying to convey to you is that the existing math logic (on the ‘independent events’ issue) is rife with contradictions. The answers I supply are not thus handicapped. This is a mathematical revelation waiting to be discovered, and I am looking to you to help me bring it to the world.
There is a saying that goes: Heroes are not born; they are cornered. Thank you for your time. Sincerely,
NOTE: Four weeks passed with no reply from Mr. Snell.

R.D. Ellison
Are we still having a discussion? I trust that you have not chosen to cease being a participant without giving notice of that decision.
As an educator and a scholar, I think you have a responsibility, in a discussion that politely challenges a theory created by others in your field, to finish responding to the questions, or concede that I have made some points. I mention this only because it has now been four weeks since I sent my last message containing one question, and I have not heard from you. I am not complaining—just trying to establish what is going on at your end. If you need more time, that is okay, but if that is the case, I would appreciate an interim reply to let me know that you are still a participant in this discussion, which I hope you will agree has interesting connotations.

J. Laurie Snell
I was under the impression that I had answered your last letter. Apparently I did not and I cannot find it now so perhaps you could send it again. Sorry Laurie

R.D. Ellison
Glad to hear that we are still talking! Here is the resubmitted question:
(In an earlier group of questions, you concurred with the premise that the roulette wheel has no memory, but did not agree that this means that the numbers it generates are independent events.) So the question is:

If it isn’t the ‘memoryless aspect’ of the roulette wheel that makes these events “independent,” then what does? Now, if you don’t believe they are necessarily independent, then we have what I would call a whole new ball game. In either case, a clarification would be helpful.

J. Laurie Snell
You wrote—
You concurred with the premise that the roulette wheel has no memory, but did not agree that this “means” that the numbers it generates are independent events.
Unfortunately, I lost the previous messages so I am not sure what we said. It certainly has to do with what the term “means” means in the above statement. I do believe the following: if a roulette system has a perceived memoryless property then the MATHEMATICAL model describing these phenomena will likely utilize the MATHEMATICAL independence assumption.
To say that machine has memory, or not, is a very different sort of statement that I, personally, don’t know what such a statement really means.

R.D. Ellison
I guess I made the mistake of assuming that you had some awareness of the classic independent events argument, which gaming experts utilize to explain why gaming systems don’t work. Basically, this is:

Each and every table result at roulette (for example) is an independent event, which is not influenced in any way by previous results. This is because the wheel has no memory, and therefore cannot remember, and react to, what last occurred at that table. This is the standard argument. Today’s questions are: do you understand this argument, and do you concur with it?

J. Laurie Snell
— You wrote:
Each and every table result at roulette (for example) is an independent event, which is not influenced in any way by previous results. This is because the wheel has no memory, and therefore cannot remember, and react to, what last occurred at that table.
— end of quote —
In the first sentence I would replace “is an independent event” and then I would agree with this sentence. The second sentence just seems to say the same thing in a different way so with the addition of perceived I would obviously agree with that also.

R.D. Ellison
Thank you for your reply. I am, however, having difficulty understanding your intent. You said that you would replace “is an independent event,” but with what? That is, what phrase would you use as a substitute? Farther in your note, you said “with the addition of perceived.” Do you mean adding that word? If so, where?

If this is getting confusing, perhaps you could write what I wrote in a way that you agree with, and we can use that as a new starting point. Either way, I need to be clear on what you believe with regard to this “independent events” issue, because my whole purpose in establishing this dialogue is to disprove the existing logic!

J. Laurie Snell
Here is my trivial modification that makes us in agreement on this point:
Each and every table result at roulette (for example) is perceived to be an independent event, which is not influenced in any way by previous results. This is the same as saying that the wheel has no memory, and therefore cannot remember, and react to, what last occurred at that table.

R.D. Ellison
I do appreciate your willingness to work with me, but the introduction of the word “perceived” raises questions as to who is doing the perceiving, and what are their credentials? This takes the premise away from clarity, and toward confusion. For us to accomplish anything, each of us have to take a firm position that leaves no doubt as to our intent. Otherwise, we’re just wordsmithing.

Also, there has to be a well-defined reason why these events are independent. You seem to want to merge that reason into the original premise, but the two are separate statements. One is the cause; the other is the effect. So, we need a firm premise, and a reason for that premise. And we need for the intent to be clear, well-defined, and unmistakable.

For the moment, all I am asking from you is a Yes or No answer. You are permitted to expand upon those answers to clarify your intent, if you think that will be helpful. Here is the (improved) premise, and the nearly-universally-accepted reason for that premise, at what is assumed to be an unbiased roulette table:

Each spin of the roulette wheel is an independent event, which is not affected by any previous decisions at that table.
Do you agree? (Yes or No.)

The reason each spin is an independent event is because the wheel has no memory, therefore, it cannot recall or react to anything that previously occurred at that table.
Do you agree? (Yes or No.)

R.D. Ellison
Are you still a participant in our ongoing discussion? I am asking because more than a week has passed since my last email message was sent to you, and I am hoping to avoid a repeat of our recent problem, where I waited weeks for a reply to a question you were not aware of.

As you may know, I am seeking someone with a math background to acknowledge perceived contradictions in an existing mathematical theory. If you are unable or unwilling to help me in this regard, could you perhaps refer me to someone who might agree to review the evidence I have accumulated, or, direct me as to how I might accomplish that?

Laurie Snell
— You wrote:
Each spin of the roulette wheel is an independent event, which is not affected by any previous decisions at that table.
Do you agree? (Yes or No.)
YES

The reason each spin is an independent event is because the wheel has no memory, therefore, it cannot recall or react to anything that previously occurred at that table.
Do you agree? (Yes or No.)
YES

R.D. Ellison
Now we may proceed to the next stage.

Question #3:
Regarding question #2 from above, is this (the wheel having no memory) the only reason that the events (referred to in question #1) are considered to be independent? Thank you.

J. Laurie Snell
The question is very confusing. The perceived memoryless property is the only reason we utilize in order to endorse calling the event independent; but independence is a vastly weaker notion than memoryless (hence there could be, and often are, other possible reasons to call events independent).

I realize that I have not been very good at answering e-mail messages. Actually I am out of town quite often and going away for another week tomorrow. I would like to take you up on your suggestion that I recommend another person to try to get at what concerns you. I have been discussing your questions with my colleague Greg Leibon who has a much better understanding than I of matters that are not simply mathematical questions. He does not like to use e-mail but he would be happy to talk with you on the phone about this. His phone number is: 603-646-XXXX.

R.D. Ellison
Thank you for the referral. Is he academically qualified to discuss this matter? I don’t mean this as a slight to you or to him, but you have dealt me a very severe triple blow: you are backing out of what I think you know (by now) is a losing argument; you are referring the matter to someone who may not have scholastic mathematical credentials (but I am not making that assumption as of yet), and you also tell me that this person eschews email, which keeps his answers from being in writing, which is vitally important for me to make any progress in what I am trying to accomplish!

I would like to offer a comment. I am surprised that someone who taught mathematics at a university doesn’t own a notebook computer to keep up on his email when away. And I am also disappointed that you don’t seem to have any academic interest in the scientific and historical aspects of what we are (or were) trying to discover together. You have seen my website. You know what my argument is; you know what’s coming. Has it not occurred to you that ten years from now, the theory I have presented will be the new reality for all mathematicians, statisticians, and gaming authors? How are you going to feel then, knowing that you walked away from a discovery so obvious, toward which you could have contributed?

Thank you for your time. I will call this person Friday, when my day schedule permits (which is another reason I think email is preferable: schedules don’t have to be synchronized).