Probability of random (in)finite graphs being isomorphic
From the comment: in the mathematical field of graph theory, the Rado graph [1], also known as the random graph or the Erdős–Rényi graph, is the unique (up to isomorphism) countable graph $R$ such that for every finite graph $G$ and every vertex $v$ of $G$, every embedding of $G − v$ as an induced subgraph of $R$ can be extended to an embedding of $G$ into $R$. As a result, the Rado graph contains all finite and countably infinite graphs as induced subgraphs.

In this note you can find other details, in particular: ... if we produce two random infinite graphs, then, with probability 1, they both have the extension property. Hence they are isomorphic (and also isomorphic to the Rado graph); in other words, any two random countably infinite graphs are isomorphic ... 

[1] Rado, Richard (1964), "Universal graphs and universal functions", Acta Arith. 9: 331–340.