Here is a discussion i find interesting at Laurance board.

Posted By: Joe Player

Date: Sunday, 9 September 2012, at 9:08 p.m.

```
There is are double 00 Interblock self-contained Roulette wheels in my neighborhood CA casinos. Some machines display the results of the last 8 numbers, some display as much as the last 200 to 400 games in aggregate format. These are the "Organic" brand as opposed to the previous "Megastar" brand.
What caught my interest was the circular graph of the wheel based on the last 200 or 300 hits. In one unnamed casino, the graph shows the actual number of hits per number, respectively. I understand that with a "00" wheel, the house edge is an immutable 5.3% (rounded). Using 300 data points and that there are 38 numbers on the wheel, then using the Law of Large Numbers, we should see the each number hit at a rate of 7.9 times (rounded), which is about 8 on an integer basis. I also understand the concept of randomness, which is that each number may differ from "7.9" or "8" times. Instead, I have consistently observed a range of "0" hits per 300 to as many as "20" hits per the same 300.
```

Posted By: Stella the StatLady

Date: Monday, 10 September 2012, at 11:42 p.m.

```
Hello Joe,
Firstoff, your headline indicates that you have some sort of question, however no question is asked in the body of your post. I am not sure what you mean by a "temporary bias", as I cannot fathom what could possibly make a roulette wheel to be sometimes biased, and other times unbiased. If you are trying to determine permanent bias, then what you are looking for is Pearson's chi-squared goodness-of-fit test.
To determine the value of your Chi-squared statistic for your particular 300-point sample, you first need 1) your mean frequency, which you have already correctly calculated as 7.9 for 300 numbers; and 2) your degrees of freedom, which for a roulette wheel is the total number of "categories", (i.e.: the total numbers on your roulette wheel) minus 1. So for your 00 roulette wheel, your degrees of freedom would be 37.
Next, you need to compute the sum of the squared differences between your observed frequencies and the mean frequency. To do this, take each of your 38 observed frequencies from your 300 point sample, and subtract 7.9 from each of those numbers, then square each of those differences, and add up all 38 of those squared differences. As a final step, simply divide your final result by 38. This is the final value of your chi-squared statistic.
Now you want to know if your 300-point sample, which is now represented by the chi-squared statistic that you just computed, is in fact biased. To find out, you need one of those large statistical tables to look up your computed statistic. That statistical table is called the Chi-Squared Distribution Table. This table consists of rows and columns, in which the y-axis represents degrees of freedom (37 for you), and the x-axis represents something called a p-value, or probability. For example, a p-value of 0.05 says there's a 5-percent probability that this chi-squared statistic would be met or exceeded randomly in nature.
Finding Chi-Squared distribution tables that go as high as 37 degrees of freedom (DofF) may be difficult to find. Fortunately, spreadsheet programs such as Microsoft Excel contain a worksheet function called CHIINV, in which you provide the DofF and the p-value, and it spits out the Chi-squared statistic for those parameters.
For your p-value, you will want to use either 0.05 (a commonly used statistical benchmark), or if you want a very high level of certainty, you can use 0.01. The CHIINV function returns 52.19 for p=0.05, and 59.89 for p=0.01. This means that if your computed chi-squared statistic for your 300-point sample reaches 52.19 or greater, you can with a high degree of certainty reject the hypothesis of "unbiasedness" and assume that you are dealing with a "permanent" biased wheel.
Sincerely,
Stella
```

Posted By: Joe Player

Date: Tuesday, 11 September 2012, at 9:31 p.m.

```
Thank you for your reply. I am looking for a critical P-value at the .005 level or better. The .05 standard is a joke because it means I have a 1 in 20 chance that my observations is due to pure randomness.
To be clear, I am not looking for a permanent bias given the advances in today's slot machines. If the game can post the results of last 200 to 400 spins, it clearly has a superior data collection abilities.
Thanks again.
```

Posted By: Joe Player

Date: Tuesday, 11 September 2012, at 9:21 p.m.

```
For some reason, the second half of my message never got posted.
What caught my attention even further was that a "slice" or a "wedge" of the wheel would be "temporarily" biased. I would consistently see that this "temporary" bias and sometimes the bias would move -- the bias could last as short as an hour to as much as 5 hours and then is gone. For example, when looking at a slice of 5 consecutive numbers on the wheel (it happens to be 5 numbers since I did most of the analysis on 5 numbers, but the "slice" could be as few as 2 to as many as 10 numbers or more), I would observe the sum of these five number would hit at a frequency in excess of "50" to as much as "63" hits or more per the most recent 300 spins, respectively. That is to say not all five-number totaling "50" or greater would be considered biased.
I will not explain how this temporary bias is created or how it goes away -- you just have to use your own powers of observation. Logic dictates that something caused these 5-number slice, on a combined basis, to hit at a rate that is higher than the average of 39 hits (rounded) per 300 numbers, respectively. Again, think of the Law of Large Numbers, it's harder for larger slices to be decidedly above the average, respectively. One possible explanation is pure randomness. I could understand that "50" hits is really not much different than say the average of "39" hits, then how do you explain "60" hits. In econometrics, this would be an example of serial correlation or "rho", i.e. the error terms are correlated. In layman terms, the robustness of the pseudo-random generator is suspect for these spins.
The math works out as followings, for betting 5 random numbers at once, the hit ratio is 1 in 7.6 (or 38 divided by 5). However, you need about a hit ratio of 1 in 7.2 (36 over 5) to overcome the house edge of 5.3% (rounded). When you are hitting at a rate of 1 in 5.5 spins, then something is clearly wrong with the machine. My longest consecutive losing streak is 30 spins -- the probabilty of striking out 30-times in a row happens about 1% of the time.
Since I have studied econometrics at the graduate level a long, long time ago, I know the correct test is looking at the P-value. Has anyone else used the P-value in their test for temporary bias during roulette. For those who understand hypothesis testing, you form the hypothesis, you collect the data, and you run the analysis and look at the P-value (and other tests as well). The hypothesis is important because some 5-number slices with over 50 hits can be decoys and not be indicative of a bias. How you define a "bias" is key.
I am NOT looking for a permanent bias because I suspect the software is programmed to detect this using the standard chi-square tests and would alert the casino that the wheel is significantly biased.
```

Posted By: Laurance Scott

Date: Tuesday, 11 September 2012, at 10:02 p.

```
The questions Joe Player pose go to the heart of bias play, and is a sensitive area that will not be discussed on public boards. Decoding the nuances and shifting of biases is extremely difficult and is - in my opinion - more of an art than a science.
For those serious about bias play, I can offer this advice: Data taken for a wheel in its "natural state" will not follow the same pattern as the data you actually play. Trying to exploit a bias situation alters the bias. Subtle changes in wheel speed, ball type and even the type of dealer spin can cause what appear to be heavily positive bias numbers to turn negative.
Then, when you leave - after you have lost a good portion of your bankroll - your numbers appear again as if by magic.
Cracking this code is one of the few deep, dark secrets that remain for Roulette.
```