I don't have a book reference, but here are some rambling words about why one might, in general, expect objects $x$ to appear with probability proportional to $\frac{1}{|\text{Aut}(x)|}$. The short version is that this is a very natural number to associate to $x$.

To start with, let $X$ be a finite set. If you wanted to pick a random element of $X$, probably you would do it uniformly, and you expect that in the absence of further structure this is what would happen "in Nature."

Now suppose $X$ comes equipped with the action of a finite group $G$, and you wanted to pick a random *orbit* of $X$. One way to do this is to pick a random element of $X$ and consider its orbit. The induced probability measure on orbits, rather than assigning each orbit equal weight, assigns each orbit a weight inversely proportional to the size of its stabilizer.

This is one way to motivate the following definition. Let $X$ be a groupoid all of whose objects have a finite automorphism group which is *tame* in the sense that the sum

$$\sum_{x \in \pi_0(X)} \frac{1}{|\text{Aut}(x)|}$$

converges (where $\pi_0(X)$ is the set of isomorphism classes of objects of $X$). The above sum is called the **groupoid cardinality** of $X$, and it induces a natural probability measure on $\pi_0(X)$ where an isomorphism class $x$ occurs with probability inversely proportional to $\text{Aut}(x)$.

One way to think about groupoid cardinality is that it is analogous to the Euler characteristic. The basic intuition is that we expect $\chi(X/G) = \frac{\chi(X)}{|G|}$ for a suitably nice group action of a finite group $G$ on a space $X$, and the one-object groupoid associated to a group $G$ can be thought of as a model of the classifying space $BG = EG/G$, where $EG$ is contractible and so in particular $\chi(EG) = 1$. For further discussion of the naturality of groupoid cardinality see this blog post.

**Example.** Let $X$ be a finite set on which a finite group $G$ acts. Form the action groupoid, whose objects are the elements of $X$ and which has a morphism $s_1 \to s_2$ labeled by $g \in G$ whenever $gs_1 = s_2$. Then the groupoid cardinality of the action groupoid is

$$\sum_{x \in \pi_0(X)} \frac{1}{|\text{Stab}(x)|} = \frac{|S|}{|G|}$$

and the induced probability measure on $\pi_0(X)$ is the one we considered above.

**Example.** Let $G = \mathbb{Z}/2\mathbb{Z}$. Then $BG \cong \mathbb{RP}^{\infty}$ has a cell decomposition with one cell in every dimension, so its Euler characteristic should be

$$1 - 1 + 1 - 1 \pm ...$$

which has, say, Cesaro sum $\frac{1}{2} = \frac{1}{|G|}$!

**Example.** Consider the groupoid of finite sets and bijections. Its groupoid cardinality is

$$\sum_{n=0}^{\infty} \frac{1}{n!} = e.$$

With respect to the corresponding probability measure, a random finite set $S$ occurs with probability $\frac{1}{e |S|!}$. The distribution of cardinalities we get this way is Poisson with mean $1$.

What kind of process produces random finite sets? One candidate is to take the fixed point set of a random permutation $\pi \in S_n$ for $n$ large, and in fact one can show that as $n \to \infty$ the distribution of $|\text{Fix}(\pi)|$ approaches a Poisson distribution with mean $1$. See, for example, this blog post for details.

More generally, the distribution of the number of $r$-cycles of a random permutation in $S_n$ as $n \to \infty$ is Poisson with mean $\frac{1}{r}$. This should have an interpretation in terms of random finite sets of $r$-cycles, and indeed it does: a collection of $n$ $r$-cycles should have an automorphism group of size $r^n n!$ because we can both permute the cycles and cyclically permute the elements in each cycle, and this recovers a Poisson distribution with mean $\frac{1}{r}$.

For a related example of a more number-theoretic flavor, one can say the same thing about irreducible factors of degree $r$ in a random monic polynomial of large degree over $\mathbb{F}_q$, except that now one has to let $q \to \infty$ as well: see this blog post for details.

**Example.** Here is another example from number theory. Recall that by the Chebotarev density theorem, the Frobenius elements associated to primes $p$ in the Galois group $G = \text{Gal}(K/\mathbb{Q})$ of a number field $K$ are equidistributed in $G$ as $p$ varies. But Frobenius elements are not elements of $G$, they are conjugacy classes, hence objects, well-defined up to isomorphism, in the action groupoid associated to the action of $G$ on itself by conjugacy. So the Chebotarev density theorem can be reinterpreted as saying that a given conjugacy class appears as a Frobenius element with probability inversely proportional to the size of its centralizer.