Gregory
Leibon/R.D. Ellison Dialogue
The following are the transcribed email messages that form the dialogue of Author R.D. Ellison and Dartmouth Math Professor Gregory Leibon, which took place from August 21, 2002, to October 22, 2002. This followed the first dialogue between Mr. Ellison and retired Math Professor J. Laurie Snell, and preceded their second dialogue. Those are documented separately at this website.
Note: To clarify who is
talking, Professor Leibon's words are indented and shown in red.
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Professor
Gregory Leibon (picture from
website): |
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Subject:
At Laurie Snell’s request Dear
R.D. Ellison, Laurie
Snell has asked me to look over your claims at http://www.ildado.com/article12.html
as well as a sequence of questions you posed him. I must admit that
at first it looked to me that questions you presented were simply a
question of semantics, and not in the realm of a mathematician at
all. However after going to the above website I realized the true
content of your questions and am quite intrigued. Please tell me if
you think the following are fair deduction to be made from your
article at http://www.ildado.com/article12.html. Claim
1: A truly accurate model for roulette will allow a given spin to be
affected by past spins. Claim
2: I have devised a method, the 3qA procedure, that gives evidence
in support of Claim 1. Claim
3: I have tried out the 3qA procedure 7500 times, and found I can
produce an expected payment of p=1.0794 per dollar gambled, as
opposed to the usual expected payoff of .9474. If
we understand these claims correctly and assume that the experiment
in Claim 3 was a controlled experiment (and in particular that you
decided on the 7500 and the 3qA strategy before performing the
experiment), then the results are EXTREMELY impressive, and worthy
of investigating further Claim 1. In fact, if I am interpreting your
experiment correctly, then your claimed result is no less that 70
standard deviations better than what would be expected! Under such
circumstances, I would be forced to accept that your Claim 1 has
content, and would be very interested in attempting to replicate the
experiment in Claim 3. I ask you the following: Question:
Would you say that your experiment was performed in a controlled
manner (at least with regard to the pair of senses mentioned above)? If
you could accurately describe to me the experiment and the protocol
used in performing it, I’d be able to more accurately judge the
evidence that supports Claim 1. Gregory Leibon |
Subject: At Laurie
Snell’s request
Date: 21 Aug 2002 20:40:35 –0400
From: R.D. Ellison (web address deleted)
To: Gregory Leibon (web address deleted)
Hello Gregory,
Thank you so much for
your note. This is exactly the kind of response I have been seeking for
quite some time. I will try to answer all your questions, and I have some
comments also:
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Laurie Snell has asked me to look over your claims at http://www.ildado.com/article12.html as well as a sequence of questions you posed him. I must admit that at first it looked to me that questions you presented were simply a question of semantics, and not in the realm of a mathematician at all. However after going to the above website I realized the true content of your questions and am quite intrigued. Please tell me if you think the following are fair deduction to be made from your article at http://www.ildado.com/article12.html. |
Just a note: the article
at my own website at this address: http://www.gamble2win.com/the_big_lie.htm
is probably more concise and to the point. I hope you can check it out.
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Claim 1: A truly accurate model for roulette will allow a given spin to be affected by past spins. |
I am not sure what you
are saying, but I do claim that results at a roulette table are influenced
by the previous results at that table, through what I call Statistical
Propensity.
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Claim 2: I have devised a method, the 3qA procedure, that gives evidence in support of Claim 1. |
Yes, it is one of several
pieces of evidence I have to support the claim above.
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Claim 3: I have tried out the 3qA procedure 7500 times, and found I can produce an expected payment of p=1.0794 per dollar gambled, as opposed to the usual expected payoff of .9474. |
I am not a mathematician,
so I don’t know if those figures are the correct translation, but the
3qA strategy gives the player a 7.94% edge over the casino (larger than
what the casino normally pays itself) in two samplings of documented
roulette spins that exceed 7500 spins, which equate to 372 sessions, or
881 bets. And I am quite sure that no other strategy for roulette could
stand up to that many trials.
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If
we understand these claims correctly and assume that the experiment
in Claim 3 was a controlled experiment (and in particular that you
decided on the 7500 and the 3qA strategy before performing the
experiment), then the results are EXTREMELY impressive, and worthy
of investigating further Claim 1. In fact, if I am interpreting your
experiment correctly, then your claimed result is no less that 70
standard deviations better than what would be expected! Under such
circumstances, I would be forced to accept that your Claim 1 has
content, and would be very interested in attempting to replicate the
experiment in Claim 3. I ask you the following: Question: Would you say that your experiment was performed in a controlled manner (at least with regard to the pair of senses mentioned above)? |
I think my experiment
would meet your definition of ‘controlled,’ because the proof is
public information. That is, if you play per the rules, and use the two
prescribed system testers as verification, you will arrive at the 7.94%
figure I mentioned earlier. These system testers are available to the
public.
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If
you could accurately describe to me the experiment and the protocol
used in performing it, I’d be able to more accurately judge the
evidence that supports Claim 1. Gregory Leibon |
I would be happy to send
you a complimentary copy of my book (which contains the strategy and
rules), along with a copy of the Verification Statistics, which are meant
to be accompanied by two published system testers, which are now in
circulation. We can also discuss perhaps the lending of those system
testers, so you can see that the numbers add up.
I hope this answers your
questions for now. Thank you so much for taking the time to look at this,
and to write (using email), and for your courtesy in writing.
R.D. Ellison
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Subject:
My mailing address --- You wrote: |
<http://www.gamble2win.com/theBigLie.htm>
is probably more concise and to the point. I hope you can check it out.
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--- end of quote --- At
this website, I think you understate a bit the results of your 3qA
experiment. The current MODEL of phenomena, like the game of
roulette, includes the HYPOTHESIS that the events are independently
determined. (Though I’m using this term as a mathematician, which
is quite distinct from how you use it. Especially judging from
comments like “And anything that moves in a predictable fashion
cannot be independent”; since, to a mathematician, independent
events behave in some of the most dramatically predictable ways
imaginable – things like the central limit theorem, weak law of
large numbers, law of iterated logarithm, etc… This is perfectly
all right with us, and, in fact, it’s what makes independently
produced events so interesting to study.) As a mathematician I am
interested in models that predict the outcomes of experiments, and
not in arguing about what the word independent should mean. Hence my
interest in your work is that your 3qA evidence suggests that this
usual MODEL is flawed, and, in particular, that the independence
HYPOTHESIS may need to be altered to more accurately capture the
results of experiments. This would be a truly incredible discovery,
and not just some semantics game. --- You wrote: |
I would be happy to send
you a complimentary copy of my book (which contains the strategy and
rules), along with a copy of the Verification Statistics, which are meant
to be accompanied by two published system testers, which are now in
circulation. We can also discuss perhaps the lending of those system
testers, so you can see that the numbers add up.
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---
end of quote --- I
look forward to seeing this material. Gregory
Leibon |
Subject: Re: At Laurie
Snell’s request
Date: 22 Aug 2002 21:18:00 –0400
From: R.D. Ellison (web address deleted)
To: Gregory Leibon (web address deleted)
Hello,
Thank you for writing. I
will put copies of the book and booklet in the mail tomorrow.
Regarding the issues of
definitions, I think that was Mr. Snell’s doing. I spent months trying
to convince him that my phrasing was the standard argument, held dear by
virtually every gaming author. But he kept challenging the basic premise,
and tried to re-write it in a way that stripped out all its meaning. So we
had to go down the road of making sure we concurred on our definitions. He
never gave me a chance to convey that I could have forwarded emails to
him, sent to me by over a dozen top-selling gaming authors, who all
concurred on the definition I offered. And/or, I could have directed him
to the exact pages of published books, where he would find the same thing.
For reference, what they all say is that table events at roulette are
independent because the wheel has no memory. That is the sum and substance
of their argument. And I’m saying that the wheel was constructed to
perform a task that simulates memory, which removes the only supporting
ledge upon which the “independent events” argument stands!
I look forward to
continuing our dialogue. You sound like you are truly interested in the
subject matter, and that is a refreshing change from what I’ve been
dealing with to date. Much obliged.
R.D. Ellison
Subject: Parcel
Date: 23 Aug 2002 16:34:42 –0400
From: R.D. Ellison (web address deleted)
To: Gregory Leibon (web address deleted)
Hello,
The parcel containing my
book and booklet was sent out this morning by priority mail. You should be
receiving it early next week.
I wanted to comment on my
last message. I try to answer all my emails each night, but having a
dial-up connection, I have only a 15-minute window to do so before I get
bumped offline. So, my attempt to give you an honest appraisal of what had
previously occurred was hastily composed and did not adequately reflect my
appreciation for Mr. Snell’s efforts to help me. I hope you will
overlook this. Thanks again for your help.
Sincerely,
R.D. Ellison
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Subject:
Re: Confirmation requested --- You wrote: |
Can you please confirm receipt?
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---
end of quote --- I
did, and thank you. I have been out of town for a while and have not
had a chance to look through the materials yet. I will get back to
you as soon as I get a chance to look through your book. --- You wrote: |
I would also like to hear
your comment on what I said at the end of the second paragraph of my
message of August 22 (regarding the accepted definition of independent
events at casino games).
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I
assume you are referring to the following --- You wrote: |
what they all say is that
table events at roulette are independent because the wheel has no memory.
That is the sum and substance of their argument. And I’m saying that the
wheel was constructed to perform a task that simulates memory, which
removes the only supporting ledge upon which the “independent events”
argument stands!
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---
end of quote --- As
I understand it, I disagree with this argument. However in your
description it is not clear what is meant by “independent
events,” so perhaps this is why we disagree. In order to eliminate
this possibility, I will now describe a mathematician’s view of
independence (also this will allow you to more easily follow the
below paragraph, where I describe my problems with the above
reasoning). A mathematician’s articulation of the independence
hypothesis: Experiments A and B are independent means that the
probabilities with which Experiment A takes on its possible outcomes
does not depend on the outcomes of Experiment B. The
memoryless property is traditionally utilized in modeling roulette
by requiring, in the model, that the outcome of the “experiment”
of spinning a roulette wheel is independent of the outcomes of
previous spins, in the above sense. This is an extremely rigid
property, and conforming to it forces phenomena to behave in
incredibly rigid ways (like the weak law of large numbers, etc.).
One should NOT think of the memoryless property as saying that we
know nothing about the outcome of an event, it is saying the
opposite: we are claiming to know the exact probabilities with which
the experiment will take on its possible outcomes – which is an
ENORMOUSLY rigid assumption. I certainly do not see any conflict
between obeying these laws (which I believe you refer as a
“simulation of memory”) and the independence hypothesis. In
fact, quite the opposite: to a mathematician, these rigid laws (the
“simulation of memory”) are born of the independence hypothesis.
Quite literally the laws of probability follow DUE to the
mathematical articulation of the independence hypothesis – not in
spite of it! What
interests me about your roulette experiment is that it suggests that
utilizing the above independence hypothesis is NOT a good way to
model roulette, which would shake the very foundation of how to
apply probability to the world. If so, your discovery of this
phenomena would rank among the greatest scientific discoveries of
all time. I admit that I am skeptical, but I sincerely hope that my
skepticism proves unfounded, and that your experiments can be
replicated under controlled circumstances (which I hope to soon have
time to implement). Sincerely, Gregory Leibon |
Subject: probability and
predestination
Date: 02 Sep 2002 21:48:51 –0400
From: R.D. Ellison (web address deleted)
To: Gregory Leibon (web address deleted)
Hello,
Thank you so much for
taking the time to articulate those thoughts. This is precisely the type
of interchange I have been hoping for. I only wish I understood
“mathspeak” a little better. Every so often a word comes up that
raises questions as to how it should be interpreted. So, let me start with
what I think I understand. Please forgive me if I misinterpret your
intent.
I’m not sure I agree
that a 1 in 38 probability represents a rigid property or assumption. If
we agree that a miscast ball is disqualified from consideration, then the
ball has to land in one of those 38 slots. Your use of the term “exact
probabilities” tends to make it sound like we can pin an event down to
one number, when in fact the entire field of 38 is understood to be
possible.
Also, I question the
sentence: “Quite literally the laws of probability follow DUE to the
mathematical articulation of the independence hypothesis – not in spite
of it.” Independence, as I understand it, means free from influence.
Conversely, probability implies predestination; that is, that a certain
behavioral pattern is expected to occur. How can an event that is
considered to be predictable also be described as “free from
influence”? The two concepts conflict.
One of the reasons I’m
raising these questions is observations made over a period spanning two
decades, which have led me to understand more clearly (than most) how
these numbers behave in large groups. One observation I’ve made is that
no even-chance proposition seems to be capable of winning as many as 30
consecutive decisions. Not once, ever. Another, is that in ANY sampling of
300 roulette spins (for example), ALL numbers will have come up at least
once. This will happen every time, without exception. From these and other
examples I could offer, I think it can be safely deduced that this
‘evening-out process’ is not merely a persistent coincidence, as the
‘independence guys’ would have us think. Instead, the behavior of
these numbers is influenced by the equivalent of a countdown that adjusts
itself with every spin of the wheel. This is the only explanation I know
of, that doesn’t lead to conflict or contradiction somewhere along the
continuum of the independence premise.
Thanks again for your
help. You are very kind, and your efforts are very much appreciated!
Sincerely, R.D. Ellison
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Subject:
Re: probability and predestination --- You wrote: |
“exact probabilities”
tends to make it sound like we can pin an event down to one number, when
in fact the entire field of 38 is understood to be possible.
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---
end of quote --- Exactly
the opposite. Exact probabilities in this setting is the claim that
EACH of the 38 possibilities is EQUALLY likely, an extremely rigid
assumption. Different real world experiments may or may not respect
such an assumption. --- You wrote: |
Independence, as I
understand it, means free from influence. Conversely, probability implies
predestination; that is, that a certain behavioral pattern is expected to
occur. How can an event that is considered to be predictable also be
described as “free from influence”? The two concepts conflict.
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---
end of quote --- No.
Independence to a mathematician, is exactly the claim in the
previous e-mail (which resembles the above claim but only in a
certain naïve sense). In particular, one assumes that they have an
understanding of the probabilities of a given experiment, regardless
of its dependence or independence of other experiments. Like ALL
events, independent events are assumed to experience the above sense
of “predestination,” and there would be no notion of probability
at all if events in the real world did not often mimic this
assumption. If you wish to familiarize yourself with the
mathematician view of these notions I highly recommend looking over
Laurie’s book at http://www.dartmouth.edu~chance/teaching_aids/books_articles/probability_book/book. --- You wrote: |
One observation I’ve
made is that no even-chance proposition seems to be capable of winning as
many as 30 consecutive decisions.
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---
end of quote --- Using
standard probabilistic results you could only rightfully hope to see
such a streak occur if you performed around 10 billion consecutive
even odds experiments! Furthermore it would be hopelessly unlikely
if you only performed say a million (then you would expect to see a
longest streak of about 17-23), to see such a streak. I’m not sure
how to evaluate your claims, since I’m not sure what magnitude of
experiments you are talking about. The expected size of a longest
streak in N experiments is “usually” in [log(N/2) /log (2) –2,
log (N/2) /log(2) +4]. So you can use your own estimate of the
number of experiments, N, that you feel you have witnessed in order
to see how long of a streak you should have expected to see. (I
found these estimates in the excellent article: Mark F. Schilling,
The Longest Run of Heads, Coll. Math. J. 21 (1990), 196-207. This
article is actually not ridiculously “mathy,” and you may enjoy
it.) --- You wrote: |
Not once, ever. Another,
is that in ANY sampling of 300 roulette spins (for example), ALL numbers
will have come up at least once. This will happen every time, without
exception.
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---
end of quote --- Using
standard independence model, the probability of this happening is
about 1 in 3000. How many experiments have you tried? You would need
to have performed more than 3000 such 300-run experiments before
such a claim would have a statistical significance. --- You wrote: |
Instead, the behavior of
these numbers is influenced by the equivalent of a countdown that adjusts
itself with every spin of the wheel.
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---
end of quote --- In
order to evaluate such a claim I would need to note the scope of the
above experiments (i.e. the approximate number of even-off
experiment N and the number M of 300-spin experiments that you have
witnessed.) If N < 10 billion or M not a fair bit more than 3000
then your claims would support rather than contradict the
independence hypothesis! Also, in the future, always provide an
estimate of your experiment sizes so that I can accurately evaluate
the experiment’s meaning. Sincerely, Gregory Leibon |
Subject: Reply to message
of Sept 3
Date: 04 Sep 2002 22:25:52 –0400
From: R.D. Ellison (web address deleted)
To: Gregory Leibon (web address deleted)
Hello,
Regarding your email
message of 9-3-02: questions, comments, and observations:
1) Thank you for directing my attention to Laurie’s book. I was not aware he was published. I have ordered a copy through the AMS (online) bookstore.
2) How can I get a copy of the article by Mark F. Schilling you mentioned?
3) Regarding my ‘probability implies predestination,’ to which you replied that it is not unusual for independent events to mimic this assumption. . .I think what I said still stands. There is a big difference between a chance copycat occurrence and a predetermined expectation.
4) One point I am trying
to convey was stated on page 5 of the book I sent to you. The second
paragraph poses the question: do you think it would be possible for a
given roulette table to fail to produce the number 5 (for example) in
twenty million spins? This is the big flaw in the independence argument:
we both know that no roulette table will ever skip over a number for that
long, don’t we? And yet, if there is not a force in the world that makes
the number 5 come up, then there would be times when it would never come
up! That ‘non-event’ would in itself be a chance occurrence!
As Frank Barstow said in
his book, Beat the Casino (page 28), ‘Dice and the wheel are
inanimate, but if their behavior were not subject to some governing force
or principle, sequences of 30 or more repeats might be commonplace, and
there could be no games like craps or roulette, because there would be no
way of figuring probabilities and odds.’
What I’m getting at, is
that we need to take this thought to the next level, where you question
things like whether probabilities CAN be assigned, in the first place, to
independent events. I agree with Mr. Barstow in saying that we can’t
expect nonconforming events to be pinned down to a precise expectation.
Please also bear in mind
that the theoretical math you’re using to challenge my points is the
same theoretical math that I claim to be at least partially invalid. But I
prefer to avoid making assumptions, and am very much enjoying our
dialogue! Thank you.
PS: Please correct any
misinterpretations I may have made.
R.D. Ellison
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Subject:
Re: Reply to message of Sept 3 --- You wrote: |
Do you think it would be
possible for a given roulette table to fail to produce the number 5 (for
example) in twenty million spins? This is the big flaw in the independence
argument: we both know that no roulette table will ever skip over a number
for that long, don’t we?
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---
end of quote --- We
do and we do not. One of us believes it is impossible and the other
believes it to be incredibly unlikely. Under the independence
hypothesis this could conceivably occur. I do not feel this is a
flaw in the hypothesis since it gives a chance of only
(1-1/38)^(20000000), which is INCREDIBLY remote! So we would all
agree that in practice for a fair roulette wheel to suffer such a
phenomena would be astonishing, though, we would phrase the
situation differently. I would say that the chances of this
occurring are so slim that we can expect that it will never occur in
the world; I would certainly NOT say that it is inconceivable (only
that it is incredibly unlikely). In practice, maybe only a
mathematician would think of (1-1/38)^(20000000) as not equal to
zero, but from a philosophical point of view this is a serious
discrepancy between your belief and utilizing the independence
hypothesis. In
part, because how do you set the cutoff between extremely rare
events that you consider impossible and extremely rare events that
you simply consider extremely rare. --- You wrote: |
where you question things
like whether probabilities CAN be assigned
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---
end of quote --- Indeed
this is a reasonable question. Much moreso in other settings (like
quantum mechanics and finance), but it has little to do with the
independence hypothesis. If you’d like to see a beautiful
introduction of how to actually analyze games WITHOUT utilizing the
conventional wisdom about probabilities, I highly recommend looking
at the introduction to Probability and Finance: It’s Only a Game!
By Shafer and Vovk at http://www.cs.rhul.ac.uk/~vovk/book/ You
might like this approach to probability better than the conventional
approach, since its mathematical articulation is much closer to
philosophical roots. It allows one to make sense of statements like
“rare events don’t occur” in a sound way. You might also like
Chapter 2 where various issues concerning the philosophical
foundations of probability are addressed in a beautifully-crafted
historical context. --- You wrote: |
Please also bear in mind
that the theoretical math you’re using to challenge my points is the
same theoretical math that I claim to be at least partially invalid.
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---
end of quote --- Perhaps
I misunderstood your argument. It sounded to me as though you were
attempting to find data that was inconsistent with this
“theoretical math,” and, in particular, with its independence
hypothesis. If you were, then I was simply pointing out that your
evidence was, in fact, completely consistent with the conventional
beliefs and tools. Hence it gives one no reason to look for
alternative tools. If you want to change the way people think about
these things you will need to replicate an experiment that actually
produces results inconsistent with those expected utilizing
conventional techniques. The data in your previous e-mail certainly
did not do this. Once
again, I am not a philosopher and I am really only interested in
discussing philosophical aspects of probability if there is DATA to
suggest that such a discussion has value. Your anecdotal data
certainly did not qualify as such, and I was merely attempting to
explain why. --- You wrote: |
How can I get a copy of
the article by Mark F. Schilling you mentioned?
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---
end of quote --- I
would be happy to mail you a copy. Please send me your mailing
address. Gregory Leibon |
Subject: contradictions
Date: 06 Sep 2002 10:58:20 –0400
From: R.D. Ellison (web address deleted)
To: Gregory Leibon (web address deleted)
Hello,
As I understand it, you
wish to avoid a philosophical debate, and concentrate on the facts. That
would be my choice as well, if it were practical. But our discussion has
already taken us to an example that would require more than a lifetime of
data accumulation. For example, my last letter posed a question for which
you offered (1-1/38)^(20000000) as the answer. I don’t know what that
works out to, but the problem is, neither of us could ever aspire to
witness such an event.
For the moment, it would
seem to be more fruitful for us to focus on the contradictions I perceive
in the existing theory. This is not exactly philosophy. It is a quest for
a logical explanation.
In my last letter, I
questioned whether probabilities can be assigned to something as random
and unstructured as ‘independent events.’ You granted that it was a
reasonable question, but I could find no answer in your reply. You also
wrote that it has little do do with the independence hypothesis, but
offered no reason for that view. I am ready for your answer now.
Regarding the
contradictions I mentioned: as I understand it, gaming authors,
statisticians and math experts alike all agree that, given a large enough
sampling, any group of unbiased numbers (that have been formally assigned
a statistical expectation) will ultimately conform to that expectation. My
question is, how do they do that in the absence of a compelling force? In
a controlled environment where a statistical certainty is involved, there
has to be a cause, and an effect. The effect is that the numbers conform
to the statistical expectation. Are you saying that there is no cause?
That this is willy-nilly random chance that conforms solely through
unabated coincidence?
Thank you for your time.
I really appreciate your efforts.
Sincerely,
R.D. Ellison
PS: I was unable to get
through to the latest website you referred me to, but I will keep trying.
Subject: contradictions
Date: 07 Sep 2002 12:16:35 –0400
From: R.D. Ellison (web address deleted)
To: Gregory Leibon (web address deleted)
Hello,
I neglected to include my
mailing address in my last message, as had been previously discussed. It
is:
R.D. Ellison
(deleted)
(deleted)
Thank you for your kind
offer to forward that article to me. L8er,
RD
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Subject:
Re: contradictions --- You wrote: |
For the moment, it would
seem to be more fruitful for us to focus on the contradictions I perceive
in the existing theory. This is not exactly philosophy. It is a quest for
a logical explanation.
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---
end of quote --- If
you want a logical explanation, this is easy enough. Any standard
probability text will show you how the laws of averages, etc. are
consistent, and in fact derivable from, a setup that assumes the
independence hypothesis. This is not a simple or intuitive thing,
but you will find it done very nicely in Laurie’s text. In
particular, there you will see that there is no contradiction in
assuming the usual independence hypothesis, though this hypothesis
is perhaps unlikely to capture your definition of independence. You
should read Laurie’s text and decide for yourself. --- You wrote: |
Regarding the
contradictions I mentioned: as I understand it, gaming authors,
statisticians and math experts alike all agree that, given a large enough
sampling, any group of unbiased numbers (that have been formally assigned
a statistical expectation) will ultimately conform to that expectation. My
question is, how do they do that in the absence of a compelling force?
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---
end of quote --- This
is highly dependent on the sense in which you mean conform. Once
again the sense in which a mathematician means this is standard, and
I cannot do it justice in an e-mail correspondence, since it is a
little involved. Fortunately, I don’t have to, since, in Chapter 8
of Laurie’s book, you can explicitly see that such a compelling
force most certainly can and does exist, in the presence of the
standard in independence hypothesis. In fact, your proposal would
probably have it conform too well. It is known that if you flip a
fair coin N times that it will be very unlikely that the number of
heads that show up, H, to be too close to N2 (of course H/N will be
near 1/2 but the H will NOT be near N/2). In fact, the number of
heads will satisfy the normal distribution (see Laurie’s Chapter
9), at least in the presence of the independence hypothesis. The
normal distribution would be unlikely to arise if one removed the
independence hypothesis, as you propose, however it is well known
that actual statistics conform very well to this normal
distribution. --- You wrote: |
Are you saying that there
is no cause? That this is willy-nilly random chance that conforms solely
through unabated coincidence?
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---
end of quote --- No.
As in the previous e-mail, this is due to the fact that we believe
there is some underlying tendency that is articulated as a
probability. For example, for a fair coin, if we assume that the
coin has the same chance of coming up heads as it does of coming up
tails when flipped, together with the independence hypothesis, then
the law of averages follows logically. This is carefully done in any
book on probability, and in particular in Laurie’s book (Chapter
8). Since you don’t like this assumption… --- You wrote: |
I questioned whether
probabilities can be assigned to something as random and unstructured as
‘independent events.’ You granted that it was a reasonable question,
but I could find no answer in your reply. You also wrote that it has
little do do with the independence hypothesis, but offered no reason for
that view. I am ready for your answer now.
|
|
---
end of quote --- Your
question is reasonable, from a certain point of view, and I invite
you to look at Vovk and Shafer’s book to see what this point of
view is. Their arguments are much too involved to discuss over
e-mail, but are very nicely expressed in their treatise. You should
decide for yourself whether the ideas there are of interest to you.
Here is another address where you can find their book (I hope this
one works). http://www.cs.rhul.ac.uk/home/vovk/book/ Gregory Leibon |
Subject: cause &
effect (reprise)
Date: 08 Sep 2002 09:52:30 –0400
From: R.D. Ellison (web address deleted)
To: Gregory Leibon (web address deleted)
Hello,
Thank you for your email
of 9-7-02. I am sorry to say, I was unable to find a straightforward
answer to any of my questions. So, I guess I will have to keep probing.
--- You wrote:
|
|
. . . though this hypothesis is perhaps unlikely to capture your definition of independence. |
And also wrote:
|
|
This is highly dependent on the sense in which you mean conform. |
I do not have personal
interpretations for the words I use. My definitions concur with whatever
is in the dictionary. In this context, the applicable definition of the
word independent is <free from influence, guidance, or control of
another or others> and for the word conform, it is <to be in accord
or agreement; to comply> These were derived from the American Heritage
College Dictionary. Do we concur on these definitions? This must be
established.
Regarding the <cause and effect> issue, I had asked if your position was that there is no <cause> to the <effect> that numbers conform to their inherent statistical expectations.
You wrote:
|
|
. . . this is due to the fact that we believe there is some underlying tendency that is articulated as a probability. |
That right there is the
heart of the issue, but you have stopped short of the mark. You need to
take your explanation at least one step further. There has to be a CAUSE
for that underlying tendency. It doesn’t just pop up out of nowhere and
exist for no reason. There has to be a driving force that makes it happen.
What is the cause of this <underlying tendency>?
I am not sure my budget
can justify the expenditures involved in these very expensive books you
defer to ($90 + shipping for the latest), because I happen to believe that
the philosophies therein are built upon a flawed premise. The fact that
you seem unable to summarize the arguments contained therein tends to
confirm that belief. So, please don’t take offense if I ask you to try
to encapsulate each argument so that I can receive something that
resembles a direct response to my queries.
Again, thank you very
much for your time. Sincerely,
R.D. Ellison
|
|
Subject:
Re: cause & effect (reprise) --- You wrote: |
Do we concur on these
definitions? This must be established.
|
|
---
end of quote --- The
American Heritage College Dictionary will certainly not do if you
wish to have a logical debate! A dictionary definition has many
distinct interpretations, and much more often than not, I find, any
argument utilizing such definitions will be wholly semantic; hence
not interesting to me (at all). If you are going to discuss these
topics with a mathematician, or hope to make logical conclusions via
the use of such definitions, you must certainly articulate the use
of your terminology. For example, utilizing the mathematician’s
definition of independence (see page 130 of Laurie’s book) the
following events are independent: Suppose we have a fair coin that
produces a head or tail when flipped. Let A be the event that the
first toss is a head and B be the event that the two tosses are the
same. The A and B are independent! Clearly the outcome of A
“influences” the outcome of B, but the probabilities of the
associated outcomes do not depend on each other, and this is the
mathematician’s definition. This is the notion of independence
that is used by statisticians when articulating the law of large
numbers, etc. . . Ask yourself: Do you think it agrees with your
definition? If not then our argument would only be about semantics
(meaning our view of these definitions) and not about the content of
our arguments. --- You wrote: |
There has to be a CAUSE
for that underlying tendency.
|
|
---
end of quote --- This
is not the realm of a mathematician, but if you need a CAUSE, then
the cause could be that there really IS some underlying tendency
that is articulated via a probability. In the world this tendency
could be borne in many ways. In terms of gaming, one has the
standard argument that it is too complicated to set up initial
conditions that allow one to predict the outcome, hence it is
“reasonable” to view all the outcomes as equally likely. This
will automatically imply (“cause”) the numbers to behave as the
laws of probability indicate, and has been very accurately confirmed
experimentally. If you control the initial conditions, then indeed
this hypothesis is no longer “reasonable” and such a model would
not be utilized. (I should say in places like quantum mechanics it
is believed that this tendency IS an actual physical phenomena, not
born simply of an inability to keep track of the appropriate initial
conditions; but in gaming situations the above justification is, I
believe, the standard one.) --- You wrote: |
So, please don’t take
offense if I ask you to try to encapsulate each argument so that I can
receive something that resembles a direct response to my queries.
|
|
---
end of quote --- Such
a debate would only be on a semantic level until we agree on a
precise use of terminology. As seen above we are nowhere even
remotely near such a state. If you want to argue semantics over
e-mail, then I suggest finding someone involved in the philosophy of
probability; this does not interest me. If you want to have a
conversation with a mathematician on these topics you will need to
see how we articulate these notions so you can point out at which
point you feel these interpretations are bad, and propose how to
modify them. Then these modifications can be implemented and
compared to experiment. Laurie’s book is free online (at the
address I gave you), and reading it might put you in a position to
have such a dialogue. Also I believe my refusal is justified since,
when I teach probability these arguments take 15 lectures to
demonstrate! Since they are standard (and available for free
online!) I refuse to communicate them over e-mail. I apologize that
Vovk’s book is so expensive; you might want to check and see if
your local library has a loan system. Also, when I point out
references which I claim state my position much clearer than I could
via e-mail communication, I mean it! These are good references
(especially Laurie’s book – which is, after all, free!), and
hope that you will look them over carefully before our next
communication. Once again my apologies that I am a mathematician and
not even a wee bit of a philosopher. Gregory Leibon |
Subject: clarifications
Date: 08 Sep 2002 16:32:57 –0400
From: R.D. Ellison (web address deleted)
To: Gregory Leibon (web address deleted)
Hello,
Thank you again for your
latest email.
I would like to start out
by saying that I appreciate your forbearance in a situation that is
probably as frustrating for you as for me. Please bear with me. I believe
we are on the verge of a mutual discovery that may benefit the scientific
community.
I want to respect your
suggestion that I partake of the reading material before we proceed, but
there are clarifications I can make at this time, that do not call for
that need. These are:
--- You wrote:
|
|
If you are going to discuss these topics with a mathematician, or hope to make logical conclusions via the use of such definitions, you must certainly articulate the use of your terminology. |
These were noted in my last letter. I will repeat:
Independent – free from
influence, guidance, or control of another or others.
Conform: to be in accord or agreement; comply.
This is what I believe
these words to mean. If you feel these definitions need to be amended or
expanded to be made suitable for a mathematical application, by all means
do so. The answer you offered, I am sorry to say, did not make sense to
me. You made an opening statement and then drew a conclusion, without
explaining its relevance, or how it was derived.
To my claim that there has to be a cause for the underlying tendency, you wrote:
In terms of gaming, one
has the standard argument that it is too complicated to set up initial
conditions that allow one to predict the outcome, hence it is
“reasonable” to view all the outcomes as equally likely.
That reply raises many
questions, and does not explain the cause. Perhaps, instead, you can
answer this question for me: is there a compelling force behind this
tendency, or not? If Yes, what is the source of this compelling force? If
No, how can it exist in the absence of a reason to exist?
By the way, I looked for
both the expensive books in the library downtown. They are not on record.
I did, however; learn that Laurie apparently wrote four other books from
1960-1966.
As I am sure you have a
demanding schedule, please take your time in replying. Thank you.
Sincerely,
R.D. Ellison
Subject: Two
breakthroughs!
Date: 08 Sep 2002 22:55:52 –0400
From: R.D. Ellison (web address deleted)
To: Gregory Leibon (web address deleted)
Hello!
We have just had two
breakthroughs! The first is that I now understand most of what you wrote.
It took many readings, but it now makes sense (to me). You are saying that
the numbers conform through man’s perception of those numbers. No no no.
MAN is merely the observer. No amount of wishing on the man’s part is
going to influence what the numbers do. It happens through the numbers
themselves. The numbers are the ones doing the work. Which brings us to
the second breakthrough: I am positive that the quantum mechanics angle is
the applicable argument. This is a physical phenomenon that is created
through the craftsmanship of the device (e.g., roulette wheel), or the
fairness of the trial environment (e.g., flip of a fair coin).
That was the stumbling
block. Do you see it now?
R.D. Ellison
|
|
Subject:
A summary of the standard
argument --- You wrote: |
You are saying that the
numbers conform through man’s perception of those numbers.
|
|
---
end of quote --- Perception
plays a role but the statement GREATLY distorts the argument. I will
now try and explain it carefully, in hopes that it will clarify the
standard position. In
order to justify me putting any more time into this I ask permission
to use any of our e-mail communications for pedagogical purposes.
If I'm going to put energy into carefully restating standard
arguments then I want to
be able to archive these debates and utilize in them in the future. Is this acceptable to you? back
to the argument... It would be much more accurate to say the numbers
conform BECAUSE of man's lack of precision
(though such a position should be restricted to a setting
that resembles the gaming world). It is conventional
in the gaming situation to view the situation as not
inherently random (one would take the opposite view in the quantum
mechanics setting), but rather as modeling a lack of precision.
A good view of the
standard working hypothesis would be something along these lines... WORKING
HYPOTHESIS: "The spinner lacks the precision necessary to make
a spin obey any laws other than
those that follow logically from
the belief that all the outcomes are equally likely with
respect to in a given spin and the outcome of any spin is
independent of the
outcomes of any other spins." Note:
this hypothesis utilizes the notion of independence I defined in our
previous correspondence (or equivalently the one from Laurie's
book), not the dictionary definition, which fits the above well,
but could lead
to meaningless semantic debate. A careful articulation of "the
belief that all the outcomes are equally likely with respect to in a
given spin and the outcome of any spin is independent of
the outcomes of any other spins" is accomplished in
Laurie's book (or any book on probability). About
the hypothesis: You are FREE to reject this hypothesis and of course
in many situations it will fail. For example, if the spinner is
utilizing a roulette wheel or a spinning technique that is rigged in
some way then this hypothesis fails. It is justified in standard
gaming situations in two steps.
The first is that it is believed, with very compelling
physical evidence, that the final position of the ball is INCREDIBLY
sensitive to the initial position
and velocity of the spin; and it is also believed that any
spinner you would choose to deal with would not put in the
effort necessary, or naturally be consistent enough, to reproduce
such sensitive initial conditions.
Secondly, via an application of Occum's razor, if the initial
conditions cannot be replicated then all the outcomes should be
viewed as equally likely with respect to a given spin, and the
outcome of any spin should be viewed as independent of
the outcomes of any other spins. As any application of
Occum's razor, this is
not a precise argument but rather a commonly accepted and wildly
beneficial vantage point. Namely utilizing the next CONCLUSION one
finds that real numbers behave as if this hypothesis is true. Note:
Since such an argument is based on Occum's razor, I have NO URGE to
argue this hypothesis unless there is DATA suggesting that the
following conclusion is false in a setting where the hypothesis
seems reasonable to me. CONCLUSION:
The laws of probability (like the law of large numbers, the central
limit theorem, etc...) can
be applied to numbers
derived from gambling
experiments that satisfy the above hypothesis. The
step from the HYPOTHESIS
to the CONCLUSION CANNOT be argued without changing accepted
definitions or abandoning logic, and I invite you once a gain to
look into Laurie's book to see how this is accomplished.
In particular, LIKE IT OR NOT, the HYPOTHESIS alone is CAUSE
enough for the number to obey the known laws and satisfy the
CONCLUSION. You can
feel free to debate the hypothesis, but any debate which does not
lead to a different conclusion (namely a testable difference in the
laws of probability that
come from the above CONCLUSION) would not interest me.
Gregory Leibon |
Subject: Breakthrough,
part two
Date: 09 Sep 2002 22:03:13 -0400
From: R.D. Ellison (web address deleted)
To: Gregory Leibon (web address deleted)
Hello,
--You wrote:
|
|
In order to justify me putting any more time into this I ask permission to use any of our e-mail communications for pedagogical purposes. |
--end of quote--
Thank you for the compliment! Permission is most definitely granted. May I also have reciprocal permission from you? This is an excellent time for us to establish this, because my last letter set the stage for a very powerful argument, which I will split between this letter and the next.
All it takes to make this effective point is to compare two roulette wheels. One of these is an $8000 perfectly-balanced professional wheel, as used in casinos, and the other is a toy that cost $39. We must add this caveat: both are capable of hitting all the playable numbers. Question: which one of these two wheels is more likely to produce numbers that conform to their inherent statistical expectations, in the long run?
This is a no-brainer: the expensive wheel will be more accurate in the long run. Do we concur? If Yes, then in the next letter, you will see very vividly what I have been trying to say all along!
Thank you.
Rick D. Ellison
|
|
Subject:
Re: Breakthrough, part two --- You wrote: |
The expensive wheel will
be more accurate in the long run. Do we concur?
|
|
---
end of quote --- I
do not. What is your reasoning and/or evidence? It seems to me
remarkable that you would find something this counter-intuitive as
to be a “no-brainer”! Would you say the same thing about
flipping a half-dollar versus a penny?! I suppose if there was a
reason it is because you have some evidence that it is difficult to
make the final holding slots on the roulette wheel of the same size,
but such a claim would need some sort of justification – it is
certainly not obvious. Without some evidence it seems the default
assumption must be that the higher quality roulette wheel will look
nicer and last longer, but will in no way produce “more random”
numbers. Any expert on chance would find any actual support of your
claim extremely interesting, and if you have any support I think it
should be published; it would probably create quite a stir among the
experts! Greg |
Subject: divergent
viewpoints
Date: 10 Sep 2002 21:14:21 -0400
From: R.D. Ellison (web address deleted)
To: Gregory Leibon (web address deleted)
Hello,
I am not believing this!
Evidently, this is where
our viewpoints diverge. You believe that the cheapest trinket available to
the market can perform as well as a precision-crafted device costing 200
times as much?
Which is to say that the
precision craftsmanship of the more expensive device is meaningless?
And, that the casinos MAY
choose the more expensive device for cosmetic reasons only?
Thank you.
R.D. Ellison
|
|
Subject:
Re: Breakthrough, part two --- You wrote: |
May I also have
reciprocal permission from you?