Let $S$ be a sphere of unit radius. Let $C_n$ be a collection of unit-radius circles/rings whose centers are (uniformly distributed) random points in $S$, and which are oriented (tilted) randomly (again, uniformly).

Q1. As $n \to \infty$, does the probability that all the rings in $C_n$ are linked together in one component approach $1$?

By "linked together" I mean that if you pick up any one ring,
all the others are physically connected and would follow.
For example, below there are $n=5$ rings, four of which are
connected, but one (topmost) is not:

Q2. Same asQ1, but with $S$ a sphere of some (perhaps large) radius $r > 1$.

Q3. Same asQ1, except with $S$ an arbitrary convex body, e.g., a cube.

I feel the answer to **Q1** should be *Yes*, but I am
less certain of **Q3**.
Exact computation of probabilities as a function of $n$ might be difficult,
but I am hoping there are relatively simple arguments to settle these
questions. Thanks for ideas!

**Answered** (*1May13*).
The combination of
Ori Gurel-Gurevich's
and
Benoît Kloeckner's postings constitute a rather complete answer,
establishing that the answer to all my questions is *Yes*, even
without the assumption that $S$ is convex.
Thanks for the interest!

Q3when $S$ is lower-dimensional (in fact, a segment is my best bet to provide a negative example toQ3). – Benoît Kloeckner Apr 27 '13 at 18:46