Summary: Let \(a_1 ,a_2\), and \(a_3\) be distinct reduced residues modulo \(q\) satisfying the congruences \(a_1^2\equiv a_2^2\equiv a_3^2(\pmod q)\). We conditionally derive an asymptotic formula, with an error term that has a power savings in \(q\), for the logarithmic density of the set of real numbers \(x\) for which \(\pi (x;q,a_1)> \pi (x;q,a_2) > \pi (x;q,a_3)\). The relationship among the \(a_i\) allows us to normalize the error terms for the \(\pi (x;q,a_i)\) in an atypical way that creates mutual independence among their distributions, and also allows for a proof technique that uses only elementary tools from probability.