  In the Art Gallery Problem we are given a polygon $P\subset [0,L]^2$ on $n$
vertices and a number $k$. We want to find a guard set $G$ of size $k$, such
that each point in $P$ is seen by a guard in $G$. Formally, a guard $g$ sees a
point $p \in P$ if the line segment $pg$ is fully contained inside the polygon
$P$. The history and practical findings indicate that irrational coordinates
are a "very rare" phenomenon. We give a theoretical explanation. Next to worst
case analysis, Smoothed Analysis gained popularity to explain the practical
performance of algorithms, even if they perform badly in the worst case. The
idea is to study the expected performance on small perturbations of the worst
input. The performance is measured in terms of the magnitude $\delta$ of the
perturbation and the input size. We consider four different models of
perturbation. We show that the expected number of bits to describe optimal
guard positions per guard is logarithmic in the input and the magnitude of the
perturbation. This shows from a theoretical perspective that rational guards
with small bit-complexity are typical. Note that describing the guard position
is the bottleneck to show NP-membership. The significance of our results is
that algebraic methods are not needed to solve the Art Gallery Problem in
typical instances. This is the first time an $\exists\mathbb{R}$-complete
problem was analyzed by Smoothed Analysis.
