OEIS A003216 gives the number of Hamiltonian graphs on $n$ points. No formula is given, but $a(8)=6196$ says there are $6196$ Hamiltonian graphs on $8$ points. OEIS A000088 gives the total number of graphs on $n$ unlabeled points. For $8$ points there are $12346$ so just over half the graphs on $8$ points are Hamiltonian. For $12$ points, the highest in the Hamiltonian list, there are $152522187830$ Hamiltonian graphs out of $165091172592$ which would claim that over $92\%$ of the $12$ point graphs are Hamiltonian.
For $n=2$ there are two graphs, neither of which is Hamiltonian. For $n \lt 8$ over half the graphs are not Hamiltonian. It doesn't seem surprising to me that once $n$ gets large most graphs are Hamiltonian. If you think about the complete graph on $n$ vertices you need very few of the edges to make a Hamiltonian path.