Does almost every pair of elements in a compact Lie group generates the connected component? It is known that almost every pair of elements in a connected compact Lie group (topologically) generates the group.
Obviously this isn't true for non-connected groups but

Given a compact Lie group $G$, is it true that almost every pair of elements of $G$ generates a subgroup containing the connected component $G^\circ$ of 1?

 A: The closed subgroups of $G$ not containing the identity component lie in countably many conjugacy classes of subgroups. So it is sufficient to show that for each closed subgroup $H$ not containing the identity component, the probability that Haar-random $g_1$ and $g_2$ both lie in some conjugate of $H$ vanishes.
Such pairs are parameterized by the manifold of triples $x \in G/H$, $g_1 \in x H x^{-1}, g_2 \in x H x^{-1}$, which is a manifold of dimension $(\dim G - \dim H) + 2 \dim H = \dim G + \dim H$. 
The image of this manifold in $G \times G$ under the projection $(x,g_1,g_2)\mapsto (g_1,g_2)$ must have measure $0$, as it is the image of a smaller-dimensional manifold (as $H$ does not contain the identity we have $\dim H < \dim G$) under a smooth map (by Sard's theorem).
For $H$ a subgroup, the same argument just barely fails to show that the probability that a single $g$ is contained in a conjugate of $H$ vanishes. Indeed in this case both manifolds have dimension $\dim(G)$. This is convenient as that statement is false, because we could take $H$ to be a maximal torus, or alternately the subgroup generated by a reflection in an infinite dihedral group, as in Noam's example. 
A: In fact, a much stronger result (due to Ito and Kawada), see Theorem 2.3 in Emmanuel Breuillard's notes. To wit, if the support of a measure is not contained in a proper closed subgroup then a random walk is eventually equidistributed. If it is contained in a proper closed subgroup, there are two cases: the first is that the subgroup is of codimension zero (in which case it contains the identity component, and we are done), or it is of positive codimension (which is obviously non-generic).
See also Stromberg 1960:
MR0114874 (22 #5692) Reviewed 
Stromberg, Karl
Probabilities on a compact group. 
Trans. Amer. Math. Soc. 94 1960 295–309. 

