For a fair game successive plays should be independent. It sounds like they are asking you to perform a test that consecutive results are uncorrelated. You could do this by pairing the data let R1, R2,...,R2n be the first 2n results. Then you can form n distnct pairs (R1,R2) (R3, R4).......(R2n-1, R2n). Calculate the pearson correlation coefficient is different from zero (if the data is continuous or even a set of integers). If the data are 0/1 for lose/win you can test for independence in the 2x2 table obtained by using the counts for (0,0) (0,1) (1,0) and (1,1). In this case of 0/1 the runs test of Wald and Wofowitz suggested above could also be used. The way it is described in the rule it sounds like they want you to construct a confidence interval for the correlation with halfwidth equal to three sigma. You would pass if 0 is contained in the interval. These tests seem to be a little too easy to pass though.

For a fair game successive plays should be independent. It sounds like they are asking you to perform a test that consecutive results are uncorrelated. You could do this by pairing the data let $R_1, R_2,...,R_{2n}$ be the first $2n$ results. Then you can form $n$ distnct pairs $(R_1,R_2)$, $(R3, R4),...,(R_{2n-1}, R_{2n})$. Calculate the pearson correlation coefficient is different from zero (if the data is continuous or even a set of integers). If the data are $0/1$ for lose/win you can test for independence in the $2\times2$ table obtained by using the counts for $(0,0), (0,1),(1,0)$ and $(1,1)$. In this case of $0/1$ the runs test of Wald and Wofowitz suggested above could also be used. The way it is described in the rule it sounds like they want you to construct a confidence interval for the correlation with halfwidth equal to $3\sigma$. You would pass if $0$ is contained in the interval. These tests seem to be a little too easy to pass though.