Probability of spurious correlation between 2 variables (eg, ratings provided by 2 independent movie-raters) depends on the range of the variables? Suppose you had two people each producing $1$ rating for a set of $n$ movies (suppose $n$ is small, $\approx 10$). Suppose the ground truth is that their ratings are randomly sampled from a discrete uniform distribution. Are their ratings more likely to be spuriously correlated if the range of possible ratings was just $\{1,2\}$ (e.g., $1$ is bad, $2$ is good) compared to the range being $\{1,2,3,4,5\}$ ($1$ is very bad, $5$ is very good)?
I ran simulations, but I would like to approach the answer mathematically.
Let $X_i$ denote movie rating $i$ by rater $1$, sampled $iid$ where $X_i \sim \text{DiscreteUnif}(1,c)$, where $c = 2$ or $5$. Similarly, let $Y_i$ denote $iid$ movie ratings $i$ by rater $2$, sampled from the same distribution as $X_i$. Our random variable of interest is the following:
$\Large R_c = \frac{\sum_{i=1}^n(X_i - E(X_i))(Y_i - E(Y_i))}{\sqrt{\sum_i(X_i - E(X_i))^2}\sqrt{\sum_i(Y_i - E(Y_i))^2}}$
Is the $P(R_{c=2} > a) > P(R_{c=5} > a)$, for any $a \in (0,1)$? Or are those probabilities equal? How do I begin formulating a solution to this problem?
 A: There's probably a more clever way to do this, but one rather naive but brutally computational is to treat $R_c$ as a function of random variables $X_i$ and $Y_i$ and compute $P(R_c>a)$ using the law of total probability and writing out the entire the sample space of every possible outcome of the variables $(X_i,Y_i)_{i=1}^n \in [c]^n\times[c]^n$ where $[c]=\{1,2,\ldots,c\}$. As we are using a uniform distribution, every outcome is equiprobable with probability $\frac{1}{c^{2n}}$. Without taking advantage of any symmetries, the computation ends up being $O(c^{2n})$. [Also, we have to assume the correlation $R$ can be defined in the case that the sample standard deviations of the $X_i$ or of the $Y_i$ is $0$, which we can just say is $0$.]
From doing this out for some very small examples where $n=3$, the probability of a spurious correlation, in the sense that $R_c>0.5$, increases as the number $c$ of options increases. The probabilities for $c=3$ to $c=10$ are 0.09375, 0.18107, 0.231445, 0.258816, 0.277649, 0.288859, 0.297638, 0.303748, 0.308406.
