There is a theorem from Erdos and Renyi that says that a random graph on $\aleph_0$ vertices (where each pair of vertices is connected with probability equal to $\frac{1}{2}$) will be isomorphic to the Rado graph with probability 1. Does an analogous result hold for random graphs on $\kappa$ vertices, for $\kappa > \aleph_0$? The proof I have read of the Erdos-Renyi theorem does not appear to generalize to larger cardinals.
More generally, how does one compute probabilities in cases like this (where, in general, the sample space and the event are both sets of the same cardinality $\lambda > 2^{\aleph_0}$. What sorts of arguments are used to show that such an event has probability $0$ or $1$. If anyone has an example of such an problem and solution, I would appreciate it!