  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.
