I am going through The Maximum Degree of a Random Graph by RIORDAN et al. On the second page, the notation $\mathbb{P}(\mathcal{D})$ is used which I assume the power set of the set $\mathcal{D}$. Right after that the quantity, $\mathbb{P}(d^{\max}(G)\le \frac{n}{2})$ is used. Here $d^{\max}(G)$ is the maximum degree of the graph $G$ of order $n$.
My question:
Shouldn't $\mathbb{P}$ always operate on sets? Why is $\mathbb{P}$ operating on integers like $d^{\max}(G)$?
As the text is about random graph, I gather that $\Bbb P(\mathcal D)$ is the probability of the event $\mathcal D$. And (as is customary) if the event is given as $\mathcal D=\{\,G\mid d^\max(G)\le n/2\,\}$ for example, we write $\Bbb P(d^\max(G)\le n/2)$ as shorthand for $\Bbb P(\{\,G\mid d^\max(G)\le n/2\,\})$.