  What is the probability that a random UHF algebra is of infinite type? What
is the probability that a random simple AI algebra has at most $k$ extremal
traces? What is the expected value of the radius of comparison of a random
Villadsen-type AH algebra? What is the probability that such an algebra is
$\mathcal{Z}$-stable? What is the probability that a random Cuntz-Krieger
algebra is purely infinite and simple, and what can be said about the
distribution of its $K$-theory? By constructing $\mathrm{C}^*$-algebras
associated with suitable random (walks on) graphs, we provide context in which
these are meaningful questions with computable answers.
