  We study how often exceptional configurations of irreducible polynomials over
finite fields occur in the context of prime number races and Chebyshev's bias.
In particular, we show that three types of biases, which we call "complete
bias", "lower order bias" and "reversed bias", occur with probability going to
zero among the family of all squarefree monic polynomials of a given degree in
$\mathbb{F}_q[x]$ as $q$, a power of a fixed prime, goes to infinity. The
bounds given improve on a previous result of Kowalski, who studied a similar
question along particular $1$-parameter families of reducible polynomials. The
tools used are the large sieve for Frobenius developed by Kowalski, an
improvement of it due to Perret-Gentil and considerations from the theory of
linear recurrence sequences and arithmetic geometry.
