**Introduction.** In 2014, MapleStory Global supplanted the existing cubing system, which featured a large selection of miracle cubes with slightly different functionalities (remember the days of the Miracle Cube, the Super Miracle Cube, the Premium Miracle Cube, the Revolutionary Miracle Cube, and the Enlightened Miracle Cube?), with a much simpler system, featuring a basic Red Cube, and Black Cube which allowed the option to choose between the existing potential and the new potential. Other differences between these cubes were a mystery at first. After a while, rumors based on empirical results surfaced, and people hypothesized that Black Cubes had a higher tier-up rate than Red Cubes. It wasn't until a while later that Nexon officially confirmed that this was true. But that doesn't necessarily tell the whole story. When Reboot world was released, one of the major benefits was the potential to collect cubing data on a much larger scale than previously possible. From various sources, it was conjectured that Black Cubes have a greater chance of yielding "prime" lines on the second and third lines than Red Cubes. But how can we determine if this is actually true? Answer: Statistics.

**Data.** The initial data set: 80 legendary potentials from Red Cubes on a pair of gloves, and 80 legendary potentials from Black Cubes on the same gloves. Each potential will be classified under one of two categories: one prime line (standard), or multiple prime lines. The Red Cubes yielded only a single multi-prime potential, whereas the Black Cubes yielded 15 multi-prime potentials. Skeptics would probably argue that this is just due to "RNGesus," and this is understandable: I certainly don't believe that the rate of multi-prime potentials is quite as low as 1/80. To supplement this data, I replicated the procedure on another pair of gloves. For the new dataset, 80 Red Cubes yielded 4 multi-prime potentials, and 80 Black Cubes yielded 19 multi-prime potentials. Convinced yet? Fortunately, statistics isn't based on drawing conclusions by eyeballing data; it's based on drawing conclusions based on the sound mathematical theory behind hypothesis testing.

Videos of the cubing are provided below so you can be assured that I'm not forging data or anything. Keep in mind that the lines "10% chance to [do shit]" are prime lines.

**Hypothesis Testing.** We have a number of options to choose from for our test. To please the skeptics, why don't we just do them all? The tests that make the most sense in our situation are the __chi-squared test of independence__, which we can perform on each of our two datasets as well as the pooled data, and the __two-sample z-test for proportions__. For the chi-squared tests, the null hypoethesis is that the type of cube used and whether or not the potential has multiple prime lines are independent, and the alternative hypothesis is that the two variables are not independent. For the z-test, the null hypothesis is that the probability of a Red Cube yielding multiple prime lines and the probability of a Black Cube yielding multiple prime lines is the same, and the alternative hypothesis is that Black Cubes yield multi-prime potentials at a different rate than Red Cubes. I'll omit the technical details; you can check them yourselves if you're interested.

**Test 1: Chi-squared test for independence with Dataset 1.** The chi-squared test statistic is equal to 13.61. Under the null hypothesis, the test statistic should follow a chi-squared distribution with 1 degree of freedom, and hence the p-value is 0.00023.

**Test 2: Chi-squared test for independence with Dataset 2.** The chi-squared test statistic is equal to 11.42. Under the null hypothesis, the test statistic should follow a chi-squared distribution with 1 degree of freedom, and hence the p-value is 0.00073.

**Test 3: Chi-squared test for independence with pooled data.** The chi-squared test statistic is equal to 24.56. Under the null hypothesis, the test statistic should follow a chi-squared distribution with 1 degree of freedom, and hence the p-value is 0.00000072.

**Test 4: Two-sample z-test for proportions with pooled data.** The z-test statistic is equal to 4.96. Under the null hypothesis, the test statistic should follow a standard normal distribution, and hence the p-value is 0.00000070.

So what exactly are the rates of multi-prime potentials for each type of cube, then? Here's the thing: we can't tell, at least not with the amount of data used here. The tests only tell us that it is ridiculously unlikely that Red Cubes and Black Cubes offer the same rate of multi-prime potentials. It doesn't tell us what the rates are, and we don't have enough data to give estimated rates with confidence. Do I believe that red cubes only give multi-prime potentials 3% of the time? No, not at all; in fact, I actually believe that red cubes give multi-prime lines at a much higher rate than 3%, but I don't believe it's close to the empirical rate of 21% with black cubes.

*Published 11/27/2016*