New Probability Tables for Roulette IV

Quote: statman

Another 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

Quote: heather

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

Quote: statman

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.

Quote: statman

The probability of a number X occurring N times in M spins of a fair roulette wheel.

Quote: statman

Dear 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.

Quote: statman

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?

Quote: statman

To 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.

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.

Games of 1000 Plays

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

Quote: statman

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

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?

Quote: statman

The history of roulette exploitation shows that they do this after they have been taken for a bundle, not before.

Quote: statman

MathExtremist:

"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.

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. ...

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."

       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
 

      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

      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

Quote: statman

Here 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..

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.

#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);
}

Quote: DJTeddyBear

Buzz is a new member?

...

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

Quote: DJTeddyBear

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.

#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);
}

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

Quote: statman

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

Quote:

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

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.

Quote: boymimbo

This 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.

Quote: DorothyGale

I 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.

Quote: statman

I'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.

Quote: My reply

My 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.

Quote: My reply to his reply

As 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.

Quote: statman

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.

Quote: MathExtremist

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.

Quote: Richard A. Epstein, "The Theory of Gambling and Statistical Logic", preface

Even 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.

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.

Quote: statman

The 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.

Quote: Murphy article

The 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.