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Quote:statmanAnother poster claimed that in the major casinos all the old imperfect

wheels have been replaced with new faultless ones. I referred to an

article entitled "Roulette Bias Exposed" by Jeff Murphy in Indian

Gaming of July 2007, p. 40, in which he describes a wheel with a

chi-square of 94.59 at Seven Feathers Hotel & Casino Resort in

Canyonville, Oregon. Considering that the chi-square of a normal wheel

is around 30, this is a highly biased wheel Link

I said that I thought that they were unlikely to be found in legal US casinos. I said that I thought a study of casinos in Caracas might be interesting. I said that a study of out-of-the-way gambling locales in the Caribbean might be interesting. I even noted that the Wizard has innuendoed at very old wheels still being in use in Europe. I have never suggested that biased Roulette wheels do not currently exist.

On the first page of your first thread, you cited Allan Wilson. Good choice; I like him. So I talked about how even back in the 1950s when he had unquestionably biased wheels to play on, he and his group got backed off and the wheels repaired or replaced. Which is exactly what I would expect to happen even if you could find a biased Roulette wheel in a legal US casino; you would not be allowed to exploit it (as I've noted in two of your other threads). They were allowed to play on the biased wheel at Harold's Club only because they attached the highest high roller who played there to their crew, and that only bought them a little more time -- the owner ended up switching the wheel out with the biggest whale he'd ever netted sitting at the table.

And (although I haven't mentioned it yet), I also think that <10K spins will always be too small a sample. Unless I've missed something, your tables aren't even based on actual Roulette spins; they were created using Excel's buggy RNG. So a wheel whose results don't match results created by a nonwheel must be biased? I'm having trouble swallowing that one.

Quote:heatherUnless I've missed something, your tables aren't even based on actual Roulette spins; they were created using Excel's buggy RNG.

And a buggy formula. It's all well and good to repeat research into roulette bias detection, but let's not start the conversation by saying "I've made a new discovery in statistics" or "I've solved a 150-year-old problem." Using Excel's built-in RNG and plotting a histogram of the outcomes doesn't qualify as a new discovery in statistics.

Biased roulette wheels exist. Basically every mechanical wheel is biased to some degree. However, it's very hard to detect that bias, certainly without many trials and excellent recordkeeping (which includes far more than just the numeric outcome). Many casinos today use bias-detection software which will discover any bias sooner than a player can, and wheel mechanisms are much more precise than they used to be. So by all means, try to detect a bias if you wish. But you don't need any RNG trials to do that. The probability of a number X occurring N times in M spins of a fair roulette wheel is directly calculable.

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Quote:statmanYou say " The probability of a number X occurring N times in M spins of a fair roulette wheel is directly calculable." I agree, but can you do it?

I'll bet you can't. I don't mean the probability of a particular number, I mean the probability of any number.

Wrong guy to challenge buddy. His name should have been your first hint

Quote:statmanThe probability of a number X occurring N times in M spins of a fair roulette wheel.

p(X occuring exactly N times in M spins) = (combin(M,N) * 37^(M-N))/38^M

--Ms. D.

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That's the weight of the binomial distribution and it gives the probability of some particular number such as 17 or 34 coming up exactly N times in M spins. In my paper I demonstrate why the binomial distribution doesn't apply to the game of roulette because it is not a simple win-lose game: it is a game with multiple outcomes. What is needed is the probability that some number or other of all of the 38 on the wheel will come up N times. It's like the probability of winning a lottery twice. The probability of a particular person doing so is very small but the probability of someone or other doing so is much greater.

I treat that in my other book on Amazon that Buzzpaff is so crazy about.

Quote:statmanDear MathExtremist,

You are referring to the program I use in Corel Quattro Pro X5 to create the simulation histograms. Where is the bug? I am calling your hand.

My probability distribution is calculated using a formula.

You say " The probability of a number X occurring N times in M spins of a fair roulette wheel is directly calculable." I agree, but can you do it?

I'll bet you can't. I don't mean the probability of a particular number, I mean the probability of any number.

Seriously? The probability of "any number" occurring N times in M spins is 1 regardless of M and N. There is no such thing as a spin which does not yield "any number" -- if the ball bounces out, it doesn't count as a spin. If you don't mean what you actually said ("any number"), then you must be interested in the probability of some particular number appearing (or number from a particular set of numbers). If that's the case, the binomial distribution absolutely applies -- either the number appears with probability p, where p = n/38 and where n = the size of the set of target numbers, or the number does not appear with probability 1-p.

Edit: I think I deciphered what you're trying to say, which is that you don't care *which* number appears N times, but just that one or more of them does. In that case, the answer is found in the multinomial distribution. You want all partitions of M into 38 buckets where at least one of the coefficients is N. But I see you made it a bet. What do I win?

The bug I referred to was in your Excel formula. You were generating numbers in the range [0..38]. You said you observed too-small frequencies for a double-zero wheel, but that's because you weren't simulating one.

But buggier still is the idea that you need to generate random numbers at all to do roulette bias detection. The roulette wheel is the random variable. Introducing another is neither necessary nor productive.

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In one of the deleted threads, I suggested that the RNG in Excel was not adequate for your job.Quote:statman3. You object to using a random number generator to simulate a roulette wheel. Do you know of a better way? Do you know a source of real roulette data?

Of course, at the time I didn't really understand what you were doing. I still don't but I've got a better idea.

It seems to me that if your goal is to discover biased roulette wheels, the system / procedure / program / whatever that you're using, will only prove that the RNG is biased or inadequate.

For what it's worth, LONG before you manage to discover a biased wheel, the casino will discover it, and fix it. Or, at the very least, move the tables around so you'd have a hard time being sure of your analysis.

TCS John Huxley manufactures roulette wheels, history displays, and a variety of other items. One of the tools they provide is wheel analysis.

http://www.tcsjohnhuxley.com/en/gaming-systems-and-security/roulette-wheel-analysis.html

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

It seems to me that, on this chart, the green dotted line is the median, the green solid lines are the standard deviation. I.E. Within the green is the safe zone. Between the green and the red line is the danger zone, and beyond the red line and you've got a bias. If those assumptions are right, the wheel in this chart is biased towards #11 and it's neighbor, #30.

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

Quote:statmanTo date I have been doing simulations only with fair wheels. Some have suggested I simulate a biased wheel. Here are some results. The hot number on this wheel is 00. Its probability of coming up is 3/76 and its expectation is +0.42105. This was arranged by taking some sample space away from No. 37 and giving to 00.

Games of 1000 Plays

This was done with a Quattro Pro spreadsheet, which I'll be glad to share with anybody. Just send me your personal e-mail address and I'll include it as an attachment. The number of plays is easily changed and the probability of any number can be changed in increments of 1/152 provided that the probability of some other number is changed the same amount in the opposite direction.

Game No. Most Hit No. of Hits 1 17 39 2 00* 59 3 9 41 4 00* 44 5 5 39 6 00* 43 7 15 41 8 6 39 9 00* 40 10 00* 49 11 6 39 12 00* 41 13 16 35 14 00* 42 15 27 36

This just shows that if you have a very biased wheel, in which one single number has a 50% more chance of happening than any other number, you still require at least 2000 spins to show some evidence of bias. Right?

Unfortunately, assuming that you have all of your statistics right, the product is just not useful.

Quote:statmanPerhaps you also will make an original contribution to the theory and win the admiration of your colleagues.

What theory? The multinomial distribution completely describes the probability of a specific distribution of roulette numbers in M spins for a fair wheel. Summing these probabilities for all sets of coefficients which meet your criteria (at least one roulette number occurring N times) yields the overall probability. Standard statistics can describe the likelihood that a sampling from a real-world wheel is also fair based on the assumption of equiprobable outcomes. This isn't novel. What exactly do you think you've discovered, invented, or solved?

As to your typo, it appears you made it in your post, not your spreadsheet. You previously wrote "RAND()*39".

Quote:3. You object to using a random number generator to simulate a roulette wheel. Do you know of a better way? Do you know a source of real roulette data?

No, I object to simulating a roulette wheel at all. It's unnecessary if your goal is to detect biases in a real physical roulette wheel. The chi-square test, for example, requires two sets of data: observed results and expected results. For roulette, you should sample the physical wheel in question to get the observed results, and you should ideally calculate the expected results. You shouldn't simulate the expected results unless you can't derive them properly. In this case, you can -- see above.

The point is that you don't need to be mucking around with virtual roulette wheels and plotting histograms and running random trials with spreadsheets. You can directly calculate F(N,M) = probability that at least one of 38 equally-likely outcomes appears N or more times in M trials.

But why would you? Are you planning on doing K samples of M-length trials and comparing each sample to F(N,M)? Why is that approach better than simply doing a chi-square test on a series of observed results vs. the expectation that each number appears with p=1/38?

This is the same argument that dice controllers use. What degree of certainty do I have that I'm actually controlling the dice versus randomness? What sample size is required to know that I'm beyond the randomness? With a sample size of 144, if I roll 12 sevens instead of the expected 24, am I controlling the dice?

Math is saying that the RNG in Excel and many other spreadsheet programs are faulty. There have been studies on this, but none on the current version, and none on QuarkExpress.

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Since you're new here, I'll direct you to one of my threads - pretty much all about A.C.

Atlantic City Casinos and Points Of Interest Map

I'd bet that much of that history was before TCS John Huxley and others came out with the sensors and history displays. After all, incorporating a histogram like the one pictured in my post above, seems like an obvious and cheap add-on once you go thru the expense of the sensor and dislplay. Hell, it might have been one of the original selling points of the history board.Quote:statmanThe history of roulette exploitation shows that they do this after they have been taken for a bundle, not before.

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Quote:statmanMathExtremist:

"You can directly calculate F(N,M) = probability that at least one of 38 equally-likely outcomes appears N or more times in M trials."

Yes, that is precisely what I do, and that is what goes into the tables, but I haven't seen anybody else do it.

That might be because it's not useful. Suppose you compute the likelihood that 100 spins yields 3 or more appearances of at least one number. Call that P. Then you watch a wheel for 100 spins. That's about 2 hours. Then what? You're comparing two hours' of results with a single probability to determine what exactly? You can't make a credible assertion that the wheel is or is not biased based on P and the past 100 spins.

So what are you getting at?

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Quote:statman... As before, 00 is the biased number with a probability of 3/76 at the expense of No. 36, which is given a probability of 1/76. All other numbers have a probability of 1/38. ...

If you have a wheel that is so outrageously biased that one number is 50% more likely to be hit than the other numbers ?????

I think this is something similar to having the slot for number 00 being 50% wider than the average slot. People would look at the wheel and say, "WTF is going on with the slot for 00?" If a wheel is THAT bad, people can probably figure it out without benefit of these marvelous tables.

Now if a wheel actually does have bias but not something so absurd as in this example, yes, I suspect it will probably take thousands of spins to establish that it is biased and by how much. Maybe I'm wrong. Suppose some shoddy casino has a really wobbly wheel that gives 1/2% or 1% bias toward a number. Which of your tables would pick that up and how many spins would it take for you to be sure?

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Quote:statman"You can directly calculate F(N,M) = probability that at least one of 38 equally-likely outcomes appears N or more times in M trials."

Yes, that is precisely what I do, and that is what goes into the tables, but I haven't seen anybody else do it. I think you understand the problem now and if you stick at it you may come up with the answer. Here's a hint: "inclusion-exclusion principle."

My 2 cents after seeing statman's tables that he kindly sent to me.

I was unable to exactly match some of his data.

So I start here.

Nope27 Blog

https://wizardofvegas.com/member/nope27/blog/

"Roulette 00 Wheel "at least 1" number NOT hitting

September 26th, 2010 at 11:21:44 am

The below table is for the probability of NOT hitting "at least 1" number in x spins

formula used from:https://wizardofodds.com/ask-the-wizard/roulette/

Wizard of Odds article

question #2 at WoO. (The Wizard also shows results from his computer simulation)

Wizards' Example: for "at least 1" number NOT hitting in 200 spins

Sum i=1 to 37 [(-1)^(i+1) × combin(38,i) × ((38-i)/38)^38] = 16.9845715651245%"

This is actually easy to set up in Excel and I verified The Wizard's and Nope27's results exactly for 200 spins.

.16794 statman results.

added: snapshot of statman's table

We should all return exact results no matter what correct formula we use. Yes?

My quick 1 million sim results shows 0.169984

group middle freq freq/100

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

33.50 <= x < 34.50 34.00 7 0.00%

34.50 <= x < 35.50 35.00 510 0.05%

35.50 <= x < 36.50 36.00 12311 1.23%

36.50 <= x < 37.50 37.00 157156 15.72%

37.50 <= x < 38.50 38.00 830016 83.00%

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

34.00 7

35.00 510

36.00 12311

37.00 157156

38.00 830016

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

cumulative

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

33.50 <= x < 34.50 34.00 7 0.00%

34.50 <= x < 35.50 35.00 517 0.05%

35.50 <= x < 36.50 36.00 12828 1.28%

36.50 <= x < 37.50 37.00 169984 17.00%

37.50 <= x < 38.50 38.00 1000000 100.00%

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

34.00 7

35.00 517

36.00 12828

37.00 169984

38.00 1000000

grouped data

items: 1000000

minimum value: 34.00

first quartile: 38.00

median: 38.00

third quartile: 38.00

maximum value: 38.00

mean value: 37.82

midrange: 36.00

range: 4.00

interquartile range: 0.00

mean abs deviation: 0.30

sample variance (n): 0.18

sample variance (n-1): 0.18

sample std dev (n): 0.42

sample std dev (n-1): 0.42

Continue please...

"at least 1" number NOT hitting in 150 spins

statman 0.50455

theoritical 0.517748

my simulation (1 million sessions) 0.517692

group middle freq freq/100

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

30.50 <= x < 31.50 31.00 1 0.00%

31.50 <= x < 32.50 32.00 13 0.00%

32.50 <= x < 33.50 33.00 234 0.02%

33.50 <= x < 34.50 34.00 2848 0.28%

34.50 <= x < 35.50 35.00 23314 2.33%

35.50 <= x < 36.50 36.00 122079 12.21%

36.50 <= x < 37.50 37.00 369203 36.92%

37.50 <= x < 38.50 38.00 482308 48.23%

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

31.00 1

32.00 13

33.00 234

34.00 2848

35.00 23314

36.00 122079

37.00 369203

38.00 482308

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

cumulative

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

30.50 <= x < 31.50 31.00 1 0.00%

31.50 <= x < 32.50 32.00 14 0.00%

32.50 <= x < 33.50 33.00 248 0.02%

33.50 <= x < 34.50 34.00 3096 0.31%

34.50 <= x < 35.50 35.00 26410 2.64%

35.50 <= x < 36.50 36.00 148489 14.85%

36.50 <= x < 37.50 37.00 517692 51.77%

37.50 <= x < 38.50 38.00 1000000 100.00%

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

31.00 1

32.00 14

33.00 248

34.00 3096

35.00 26410

36.00 148489

37.00 517692

38.00 1000000

grouped data

items: 1000000

minimum value: 31.00

first quartile: 37.00

median: 37.00

third quartile: 38.00

maximum value: 38.00

mean value: 37.30

midrange: 34.50

range: 7.00

interquartile range: 1.00

mean abs deviation: 0.67

sample variance (n): 0.63

sample variance (n-1): 0.63

sample std dev (n): 0.80

sample std dev (n-1): 0.80

"at least 1" number NOT hitting in 100 spins

statman 0.93518

theoritical 0.953397

my simulation (1 million sessions) 0.953301

group middle freq freq/100

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

26.50 <= x < 27.50 27.00 2 0.00%

27.50 <= x < 28.50 28.00 8 0.00%

28.50 <= x < 29.50 29.00 149 0.01%

29.50 <= x < 30.50 30.00 925 0.09%

30.50 <= x < 31.50 31.00 5115 0.51%

31.50 <= x < 32.50 32.00 21848 2.18%

32.50 <= x < 33.50 33.00 69154 6.92%

33.50 <= x < 34.50 34.00 159213 15.92%

34.50 <= x < 35.50 35.00 256162 25.62%

35.50 <= x < 36.50 36.00 271879 27.19%

36.50 <= x < 37.50 37.00 168846 16.88%

37.50 <= x < 38.50 38.00 46699 4.67%

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

27.00 2

28.00 8

29.00 149

30.00 925

31.00 5115

32.00 21848

33.00 69154

34.00 159213

35.00 256162

36.00 271879

37.00 168846

38.00 46699

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

cumulative

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

26.50 <= x < 27.50 27.00 2 0.00%

27.50 <= x < 28.50 28.00 10 0.00%

28.50 <= x < 29.50 29.00 159 0.02%

29.50 <= x < 30.50 30.00 1084 0.11%

30.50 <= x < 31.50 31.00 6199 0.62%

31.50 <= x < 32.50 32.00 28047 2.80%

32.50 <= x < 33.50 33.00 97201 9.72%

33.50 <= x < 34.50 34.00 256414 25.64%

34.50 <= x < 35.50 35.00 512576 51.26%

35.50 <= x < 36.50 36.00 784455 78.45%

36.50 <= x < 37.50 37.00 953301 95.33%

37.50 <= x < 38.50 38.00 1000000 100.00%

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

27.00 2

28.00 10

29.00 159

30.00 1084

31.00 6199

32.00 28047

33.00 97201

34.00 256414

35.00 512576

36.00 784455

37.00 953301

38.00 1000000

grouped data

items: 1000000

minimum value: 27.00

first quartile: 34.00

median: 35.00

third quartile: 36.00

maximum value: 38.00

mean value: 35.36

midrange: 32.50

range: 11.00

interquartile range: 2.00

mean abs deviation: 1.15

sample variance (n): 1.98

sample variance (n-1): 1.98

sample std dev (n): 1.41

sample std dev (n-1): 1.41

I do not have or have seen statmans formula for calculating at least 1 or more in N spins. I can only run simulations at this point in time.

150 spins at least 1# with exactly 9 hits

.371208 sim results

.36028 statman results.

150 spins at least 1# with exactly 5 hits exactly

.998709 sim results

.99843 statman results.

We are getting closer but I have a feeling that a formula(s) somewhere is in error.

Probably with PIE (principle of inclusion-exclusion) I love chocolate pie.

This is a good math exercise for any that want the challenge.

Statman did a good job for what I have seen so far.

I just could not verify his results at this time.

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Buzz is a new member?Quote:statmanHere is a message addressed directly to the Wizard on his private e-mail.

Dear Shack,

I would like to draw your attention to the thread "New Probability Tables for Roulette IV" in the Free Speech Zone of your forum and recommend that you expel buzzpaff from the forums. He is a relatively new member and has shown himself to be an ignorant and malicious bully.

If, having made a thorough study of the material presented there, you agree with buzzpaff, please delete the entire thread..

buzzpaff

Member since: Mar 8, 2011

Threads: 56

Posts: 1441

statman

Member since: Sep 25, 2011

Threads: 5

Posts: 27

You're here four days and you've caused more ruckus than Buzz has in his 6.5 months!

Quote:statmanThe Pr that at least one of the numbers 0 through 37 will not be hit by the 200th spin can be expressed

as 1 - (1 - (37/38)^200)^38. This evaluates to 0.16793785. The Wizard gets 0.1698457.

Simulation of 10M experiments where 200 spins are performed in each experiment ...

Percentage of times 1 or more numbers un-hit: 0.169852

#include <stdio.h>

#include <time.h>

main() {

int c1, c2, i;

int hits[38] = {}; // 37 = 00

double r = 0;

srand(time(NULL));

for (c1 = 0; c1 < 10000000; c1++) {

i = 0;

for (c2 = 0; c2 < 38; c2++)

hits[c2] = 0;

for (c2 = 0; c2 < 200; c2++)

hits[rand()%38]++;

for (c2 = 0; c2 < 38; c2++) {

if (hits[c2] == 0) {

i = 1;

break;

}

}

if (i == 1)

r++;

}

printf("Percentage of times 1 or more numbers un-hit: %1.6f\n", r/10000000);

}

This code is easy enough to write, even in Fortran, that one wonders why Statman did not check his results by a larger simulation.

Edit: Since I posted this, just moments ago, Statman has edited all of his posts saying "goodbye" ... I assume he is embarrassed at how wrong he was after the guys & gals looked at this work ... because he is gone, let me editorialize and say he was one of the most arrogant and unpleasant voices on this site in quite a while. His "know it all" pedestal grew very tiring ... always talking down to those who clearly have more expertise, education and insight than he does ... thanks Guido for taking the time and skill to open the floodgate ...

--Ms. D.

FYI: In a private message, statman took exception to these comments. I believe my reply didn't please him, prompting him to remove his posts.Quote:DJTeddyBearBuzz is a new member?

...

You're here four days and you've caused more ruckus than Buzz has in his 6.5 months!

FWIW: I think it was a mistake to remove the posts.

I don't understand why he used you as an excuse to leave ... he didn't use you as an excuse to arrive ... silly, childish & irrational person ...Quote:DJTeddyBearFYI: In a private message, statman took exception to these comments. I believe my reply didn't please him, prompting him to remove his posts.

I got the bug to finish the code I wrote ... for the record:

#include <stdio.h>

#include <time.h>

main() {

int c1, c2, x;

int hits[38] = {}; // 37 = 00

double r[38] = {};

double c[38] = {};

srand(time(NULL));

for (c1 = 0; c1 < 10000000; c1++) {

x = 0;

for (c2 = 0; c2 < 38; c2++)

hits[c2] = 0;

for (c2 = 0; c2 < 200; c2++)

hits[rand()%38]++;

for (c2 = 0; c2 < 38; c2++) {

if (hits[c2] == 0) {

x++;

}

}

r[x]++;

}

for (c1 = 0; c1 < 38; c1++)

for (c2 = c1; c2 < 38; c2++)

c[c1] += r[c2];

printf("10 million trials of 200 spins ...\n");

printf("----------------------------------\n");

for (c2 = 0; c2 < 38; c2++)

printf("%2d %1.6f %1.6f\n", c2, r[c2]/10000000, c[c2]/10000000);

}

Output -- 10M rounds of 200 spins each, with a bit of editing to make the output clearer ...

p = probability of not hitting that exact number in 200 spins

q = probability of not hitting that number or more in 200 spins

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

# not hit p q

0 0.830074 1.000000

1 0.156806 0.169926

2 0.012547 0.013119

3 0.000557 0.000572

4 0.000015 0.000015

5 0.000001 0.000001

6 0.000000 0.000000

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Quote:statman3. DorothyGale, who despite what she thinks of me, seems to be a competent C or C++ programmer.

You are in NO position to judge my skills and compliments from you aren't worth the electrons they're written with ... you couldn't program yourself out of a wet paper bag ... you couldn't even be bothered to double check your own results ... that program took me the better of 4 minutes to write and 90 seconds to execute ... by the way, it's a C program.

Quote:Here is what I think is some worthwhile reading from William Feller of Princeton University

This is just a brief discussion of inclusion/exclusion, which appears in countless books ...

Edit. Once again Statman deleted his post after my reply ... what a guy ...

--Ms. D.

Great work.

Quote:statman

The Pr that at least one of the numbers 0 through 37 will not be hit by the 200th spin can be expressed

as 1 - (1 - (37/38)^200)^38. This evaluates to 0.16793785. The Wizard gets 0.1698457.

Actually, that's not the probability that "at least one of the numbers will not be hit by the 200th spin". Here's how to break it down:

A. (37/38)^200 = probability that one specific number is absent for 200 spins

B. 1 - A = probability that A doesn't happen, i.e. the probability that one specific number appears at least once in 200 spins.

C. B^38 = probability that B (not-A) happens 38 times in a row, i.e. the probability that one specific number appears at least once in 200 spins 38 different times (38 sessions of 200 spins).

D. 1 - C = probability that C doesn't happen, i.e. the probability that one specific number does not appear at least once in 200 spins 38 different times.

The probability that "one specific number does not appear at least once every 200 spins for 38 sets of spins"

is not the same as the probability that

"at least one number does not appear over 200 spins".

So statman had perhaps the right answer to the wrong question. Regardless, neither question is relevant to the reliable detection of a biased wheel over 200 spins. You can't tell anything with one datapoint.

I think he had no idea who his audience was ...Quote:boymimboThis is a prime example of why I love this site. Someone called statman comes on and claims he's found something that hasn't been found in 150 years, provides the program. Buzzpaff flags, MathExtremists debunks, DorothyGale finds the bug in the program, debunks, and this guy deletes all of his posts.

Great work.

BTW, Guido was the 111! He made it all happen by requesting the data.

And MathE actually told him the solution early on, the multinomial distribution ... I actually looked that one up (I forgot about it) after M.E. said it ... bingo ... then he suggested summing over the partitions of M not containing N ... Yikes ... but it's the perfect solution. Ramanujan, where are you?

--Ms. D.

Quote:DorothyGaleI think he had no idea who his audience was ...

BTW, Guido was the 111! He made it all happen by requesting the data.

And MathE actually told him the solution early on, the multinomial distribution ... I actually looked that one up (I forgot about it) after M.E. said it ... bingo ... then he suggested summing over the partitions of M not containing N ... Yikes ... but it's the perfect solution. Ramanujan, where are you?

--Ms. D.

I am sure Statman learned something of lasting value.

I even sent him my Excel worksheet with the Wizards formula already set out.

So in the beginning it was the "Ask the Wizard" article and his excellent reply.

I remember when I tried to first duplicate the Wizard's answer, I came up with what MathEx described and exactly what the Wiz mentioned not to do in his article.

To do what Statman wants to do in a spreadsheet takes many hours, even with some VBA.

Would be faster to have a program run it.

But then, as ME pointed out, the results really do not have any true value in finding biased roulette wheels, other than saying I did the math correctly.

Some of it eventually may reappear on the web site of the

Rancocas Valley Journal of Applied Mathematics

Please flag this page so that it may be deleted.

Many thanks to those who have been helpful.

Of course you shouldn't claim anyone else had wrong results, because they weren't ... your results were wrong ... period. No round off error, no cancellation error, no double-precision error, no Autocad , no Maple, no compute algebra system -- just wrong. Your methods are wrong ... your results are wrong ... what a bunch of nonsense to try and blame it on something else when the blame is squarely on your lack of mathematical and programming skill ... accurate to 3 decimals is wrong ... not to mention how wrong your idea is from the start ... your conception that bias can be detected and exploited by your "method" is wrong ...Quote:statmanI'm sorry Shack didn't participate in this discussion - he could have been a great help. I went to him first. I think guido111 established that my tables are accurate at least to three significant figures, which is all you need for tests such as this.

I'm not claiming that anyone's results are wrong, but one thing you have to be on the watch for in experimental mathematics is rounding error and cancellation error. In using a spreadsheet or a compiler you will be computing with hardware double-precision floating point math. Single-precision hardware floating point math is used by vector drawing programs such as Autocad.

You will notice that in Shack's answer to the roulette no-hit problem he uses an alternating series. That is a situation where you have to be especially careful. The safest thing to do in such a case is to add the positive and negative terms separately and then compare them. Then you can tell how many places in your answer are good. Another way to avoid these demons is to use a a computer algebra system with exact math. I have been using Derive for twenty years. I still use it, but I also use Maple, which has good programming capabilities. Answers often come up in the form of a rational number: an integer of hundreds of digits divided by another integer of hundreds of digits, but it is exact.

I did post tables of simulated games with 00 rigged to have a strong favorable bias. One set was for 1,000 spins and the other was for 200 spins. In both cases the tables found the biased number three times out of four and did not return any false positives. Rude responses caused me to delete this material and everything else and it will not be back. Remember what Jesus said about pearls in the sermon on the mount? I think the tables have been shown both to be useful and to be accurate enough to remain on the bookshelves. Perhaps I shall be able to improve them. Perhaps someone will be able to verify them more precisely.

Then you have the nerve to lecture us on methodology as if you have a clue ... and then you have the cowardice to delete all of your faulty logic and talking down to experts from the site ... a little humility, somewhere, would get you started in the right direction ...

Rude responses caused you to delete? Really? They reached through the screen and pushed the delete button? No, you chose to delete ... it was your choice ... you made the decision fully and on your own ... no one here caused you to do anything ... take some responsibility for your actions, come on, be an adult ...

I'm quoting you in full from now on ...

You are a piece of work ...

--Ms. D.

Therefore, I'd like to clear the air and post my replies.

It was in his first private message where he took exception to my observation in this post, that he has been here 4 days while Buzz has been here 6 months.

Quote:My replyMy comment was not meant to be hostile. Just a statement of fact.

I stand by my statement that as a four day member, you're in no position to make the kinds of demands you're making.

I also stand by my statement in the thread you started about Buzz where I thought that Buzz is not the greatest member, but entitled to his opinion. On that note, you're also entitled to discuss your ideas. As such, I recommended the Free Speech Zone as a method to prevent it from being deleted.

That does NOT mean I support your actions or ideas. I merely support your right to have those ideas, and to present them without being deleted because they are unpopular.

It was after I sent that message, and before I saw his next PM, that he deleted his posts.

Quote:My reply to his replyAs I think about it, I realize that his [Buzz'] use of the term "thief" was inaccurate. Given his position on the subject, I think a more appropriate description would be "snake oil salesman"

Sorry to see that you're leaving, but buzz was not the only one who disagreed with your ideas. But he was one of a few who disagreed with your right to present them. As you noticed, there were many who disagreed with the ideas but who also discussed them. Thats what the forum is all about.

For the record, I'd love to know who got your threads killed. That was unfair to you.

He sent a reply to that message. I did not bother to continue the conversation.

Quote:statmanI'm not claiming that anyone's results are wrong, but one thing you have to be on the watch for in experimental mathematics is rounding error and cancellation error. In using a spreadsheet or a compiler you will be computing with hardware double-precision floating point math.

Not if you're using integers. You don't need floating-point math until the very end, and the results will be exact until then.

And your results *were* wrong because you used the wrong methodology. See my prior post that sets out how you were answering a different question than you asked. Being off by more than one percent is not rounding error.

If you know how to use Maple, it has the built-in function "multinomial(n, k1, k2, ..., km)". Your tables *should* be the results of a multinomial summation for n=trial size, m = 38, where at least one of the ki is zero (for the number that doesn't appear), or alternately at least one of the ki is K for the number repeated K times. If they're not, your tables are answering a different question than you think they are.

But as before, such tables aren't really useful in a casino setting. If you know the probability P of having a K-repeated number in n spins of a fair wheel, and you observe n spins of a real wheel, you have exactly one datapoint. Then what? You can't reasonably conclude anything from that experiment and it will take you hours to days to complete. You can't get away with sitting at a roulette table for a day without betting, so what are you going to do -- camp out in the coffee shop with a telephoto lens aimed at the recent-number display? That'll get you arrested.

The "Pentium Bug" is long gone ... accuracy of floating point division is precise, except for possibly the last digit ... for example, 0.000001 may be actually stored as 0.000000994 (though you won't find a 6 decimal floating point any more ... with 8 byte doubles, they're usually good to 15 digits) ...Quote:MathExtremistNot if you're using integers. You don't need floating-point math until the very end, and the results will be exact until then.

--Ms. D.

Quote:Richard A. Epstein, "The Theory of Gambling and Statistical Logic", prefaceEven mathematicians have fallen prey to the clever casuistry of gambling fallacies. Special wards in lunatic asylums could well be populated with mathematicians who have attempted to predict random events from finite data samples.

As for me not being the best poster, that is to be expected. I just wish more viewers would post. I see some post have 9 replies and 200 or 300 views. I am constantly amazed at the knowledge some people possess and take it for granted everybody else already knows it. The one thing I can say about my posts is often I get replies from people with only 4 or 5 prior posts.

PM's are always available for anyone who's feathers I have ruffled.

He removed his posts I believe, because lots of these threads show up on a Google search, and he might lose future victims !

I hope I get that credit. I think it was the right thing to do.

Betting systems: Methods of varying bet size, based on previous wins and losses, not only can't overcome the house edge, they can't even dent it. However, if you're one of the many mathematically ignorant gamblers who think adding up negative numbers can result in a positive one, please keep your comments restricted to the betting systems sty. Betting systems may not be offered for sale anywhere on the site.

There is a reason the WIZ has that rule.

You did. But the way he worded that response, caused me to think of "Illuminati".Quote:buzzpaff"For the record, I'd love to know who got your threads killed. That was unfair to you."

I hope I get that credit. I think it was the right thing to do.

But I still think it was wrong to delete them. Controvercial or not, they caused a lot of discussion. And a lot of the replies had good information in it.

Similarly, this thread has good information too. But because of the holes, not many people will understand it.

Some of it eventually may reappear on the web site of the

Rancocas Valley Journal of Applied Mathematics

Please flag this page so that it may be deleted.

Many thanks to those who have been helpful.

Quote:statmanThe chi-square test tests the wheel as a whole. The tables are intended to assess the significance of the most frequently occurring number. A high value of chi-square indicates a biased wheel. Although I haven't gone into it in detail, I suspect that a wheel may have a high value of chi-square without having a number of probability exceeding 1/35, which is what you need to make money. Casinos relying on chi-square therefore may take wheels out of service as money-losers to sophisticated players when really they are not.

That's the whole point. The casino detects bias faster than you can, and removes/corrects biased wheel problems before the player can figure it out. The Murphy article says that a wheel played over 20k spins was found to have a bias, but it doesn't say that any player detected it. He said:

Quote:Murphy articleThe duration of time the roulette inquiry took, to reach 20,504 spins was approximately 48 days. The roulette table was opened approximately 12 hours a day, with steady play estimated at around 8 hours each day.

But he then goes on to assume that a player at the *beginning* of those 48 days already knew about the bias and was able to exploit it. How much sense does that make?

But you seem to be saying that you can watch a roulette wheel for merely 1% of the time he did and reliably detect a bias (after 200 spins). Do you actually believe this? What 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?