# Concepts 6: Check for a Biased Wheel, or a Dealer’s biased spin with Chi Square Tests

Chi2

The Chi2 distribution tests for the difference between the observed and the expected in terms of frequencies.  We can apply this to a simple example:

In double-zero roulette, we have 38 numbers and the expected probability of any one of these numbers appearing is 1/38 or 0.026316 or 2.6316%.  Now, assuming you track all the results that appear on your roulette tables, you’d be able to check for biased wheels or even if your dealers have developed the muscle memory to spin at a regular area of the wheel.

As with all things probable, do note that nothing is impossible.  It may be unlikely, but never impossible.  Always correlate your findings with footage from surveillance.

 Number Probability 0-0 0.026316 0 0.026316 1 0.026316 2 0.026316 3 0.026316 4 0.026316 5 0.026316 6 0.026316 7 0.026316 8 0.026316 9 0.026316 10 0.026316 11 0.026316 12 0.026316 13 0.026316 14 0.026316 15 0.026316 16 0.026316 17 0.026316 18 0.026316 19 0.026316 20 0.026316 21 0.026316 22 0.026316 23 0.026316 24 0.026316 25 0.026316 26 0.026316 27 0.026316 28 0.026316 29 0.026316 30 0.026316 31 0.026316 32 0.026316 33 0.026316 34 0.026316 35 0.026316 36 0.026316

Let’s say we have tracked 1,000 spins on a particular roulette table.  We are thus expecting that each number would have appeared 1,000 x 0.026316 = 26.316 times.  Do note that you would have to give or take an allowance of -1σ to 1σ based on the central limit theorem.

Here are our results based on 1,000 spins:

 Number Probability Expected Occurrence 00 0.0263 26.3158 13 0 0.0263 26.3158 32 1 0.0263 26.3158 32 2 0.0263 26.3158 27 3 0.0263 26.3158 39 4 0.0263 26.3158 20 5 0.0263 26.3158 33 6 0.0263 26.3158 23 7 0.0263 26.3158 10 8 0.0263 26.3158 36 9 0.0263 26.3158 29 10 0.0263 26.3158 17 11 0.0263 26.3158 38 12 0.0263 26.3158 14 13 0.0263 26.3158 11 14 0.0263 26.3158 25 15 0.0263 26.3158 20 16 0.0263 26.3158 16 17 0.0263 26.3158 11 18 0.0263 26.3158 12 19 0.0263 26.3158 28 20 0.0263 26.3158 17 21 0.0263 26.3158 45 22 0.0263 26.3158 24 23 0.0263 26.3158 43 24 0.0263 26.3158 25 25 0.0263 26.3158 10 26 0.0263 26.3158 21 27 0.0263 26.3158 43 28 0.0263 26.3158 23 29 0.0263 26.3158 42 30 0.0263 26.3158 30 31 0.0263 26.3158 18 32 0.0263 26.3158 45 33 0.0263 26.3158 40 34 0.0263 26.3158 41 35 0.0263 26.3158 17 36 0.0263 26.3158 30

To find the Chi2 value, the following formula applies:

Sum of all (observed values – expected values)2 / expected values

This works out to be the following:

 Number Probability Expected Occurrence Observed – Expected Observed – Expected2 Observed – Expected2/Expected Sum 1 0.03 26.32 13 -13.32 177.31 6.74 172.984 1 0.03 26.32 32 5.68 32.31 1.23 1 0.03 26.32 32 5.68 32.31 1.23 2 0.03 26.32 27 0.68 0.47 0.02 3 0.03 26.32 39 12.68 160.89 6.11 4 0.03 26.32 20 -6.32 39.89 1.52 5 0.03 26.32 33 6.68 44.68 1.70 6 0.03 26.32 23 -3.32 10.99 0.42 7 0.03 26.32 10 -16.32 266.20 10.12 8 0.03 26.32 36 9.68 93.78 3.56 9 0.03 26.32 29 2.68 7.20 0.27 10 0.03 26.32 17 -9.32 86.78 3.30 11 0.03 26.32 38 11.68 136.52 5.19 12 0.03 26.32 14 -12.32 151.68 5.76 13 0.03 26.32 11 -15.32 234.57 8.91 14 0.03 26.32 25 -1.32 1.73 0.07 15 0.03 26.32 20 -6.32 39.89 1.52 16 0.03 26.32 16 -10.32 106.42 4.04 17 0.03 26.32 11 -15.32 234.57 8.91 18 0.03 26.32 12 -14.32 204.94 7.79 19 0.03 26.32 28 1.68 2.84 0.11 20 0.03 26.32 17 -9.32 86.78 3.30 21 0.03 26.32 45 18.68 349.10 13.27 22 0.03 26.32 24 -2.32 5.36 0.20 23 0.03 26.32 43 16.68 278.36 10.58 24 0.03 26.32 25 -1.32 1.73 0.07 25 0.03 26.32 10 -16.32 266.20 10.12 26 0.03 26.32 21 -5.32 28.26 1.07 27 0.03 26.32 43 16.68 278.36 10.58 28 0.03 26.32 23 -3.32 10.99 0.42 29 0.03 26.32 42 15.68 245.99 9.35 30 0.03 26.32 30 3.68 13.57 0.52 31 0.03 26.32 18 -8.32 69.15 2.63 32 0.03 26.32 45 18.68 349.10 13.27 33 0.03 26.32 40 13.68 187.26 7.12 34 0.03 26.32 41 14.68 215.63 8.19 35 0.03 26.32 17 -9.32 86.78 3.30 36 0.03 26.32 30 3.68 13.57 0.52

So, our Chi2 is 172.984.

We now look for this figure on the chi2 table.  The numbers at the top are the probability percentages and the column on the left marked df just refer to the number of categories you are looking at -1.  In our case, it’ll be 38-1 or 37.  We’ll look at df=40 as it is the closest to 37.

Our chi value of 172.984 exceeds all the probabilities on the chi2 table, which means that there is DEFINITELY a PROBABILITY of the wheel or the dealer being or doing something out of the ordinary.

So, which ones?

Residuals

We can look at each number from 00 – 36 and identify which ones were out of the ordinary.  We do this by calculating what is called a residual.  This is just fancy for how different each result is from the expected.  Here’s the formula:

Observed result – Expected result / Square root (Expected result)

So, for 00, the formula would translate to:

13 – 26.32 / square root(26.32) = -13.32/5.129 = -2.59573

 Number Probability Expected Occurrence Observed – Expected Residual 0-0 0.03 26.32 13 -13.32 -2.59573 0 0.03 26.32 32 5.68 1.108057 1 0.03 26.32 32 5.68 1.108057 2 0.03 26.32 27 0.68 0.133377 3 0.03 26.32 39 12.68 2.472608 4 0.03 26.32 20 -6.32 -1.23117 5 0.03 26.32 33 6.68 1.302993 6 0.03 26.32 23 -3.32 -0.64637 7 0.03 26.32 10 -16.32 -3.18053 8 0.03 26.32 36 9.68 1.8878 9 0.03 26.32 29 2.68 0.523249 10 0.03 26.32 17 -9.32 -1.81598 11 0.03 26.32 38 11.68 2.277672 12 0.03 26.32 14 -12.32 -2.40079 13 0.03 26.32 11 -15.32 -2.9856 14 0.03 26.32 25 -1.32 -0.25649 15 0.03 26.32 20 -6.32 -1.23117 16 0.03 26.32 16 -10.32 -2.01092 17 0.03 26.32 11 -15.32 -2.9856 18 0.03 26.32 12 -14.32 -2.79066 19 0.03 26.32 28 1.68 0.328313 20 0.03 26.32 17 -9.32 -1.81598 21 0.03 26.32 45 18.68 3.642223 22 0.03 26.32 24 -2.32 -0.45143 23 0.03 26.32 43 16.68 3.252351 24 0.03 26.32 25 -1.32 -0.25649 25 0.03 26.32 10 -16.32 -3.18053 26 0.03 26.32 21 -5.32 -1.03624 27 0.03 26.32 43 16.68 3.252351 28 0.03 26.32 23 -3.32 -0.64637 29 0.03 26.32 42 15.68 3.057415 30 0.03 26.32 30 3.68 0.718185 31 0.03 26.32 18 -8.32 -1.62105 32 0.03 26.32 45 18.68 3.642223 33 0.03 26.32 40 13.68 2.667544 34 0.03 26.32 41 14.68 2.86248 35 0.03 26.32 17 -9.32 -1.81598 36 0.03 26.32 30 3.68 0.718185

All the residuals are now in terms of σ!  If you recall, the central limit theorem infers that all data should fall within the -3σ to 3σ region in relation to the mean.

The majority of outcomes would occur within the -1σ to 1σ region.  So a reading of -2.59573 means that the number 00 has been occurring less times than expected.

Conversely, a reading of 3.642223 for number 32, indicates that the number 32 has occurred 3.642223σs more than the mean.  Definitely worth some time investigating.