If I look at the distribution of the number of small subgraphs in the random graph isomorphic to a connected graph $H$, this is asymptotically Poisson.

What proportion of these small subgraphs intersect another, fixed subgraph?

Also, what if I want them to intersect in a specific way, not just arbitrarily? Do the subgraphs occur on vertices that are selected uniformly at random, so the `striking' probability is just the probability I select those vertex, chosen uniformly at random?