In Jeffrey Weeks book "The Shape of Space" he explaines at the end of Chapter 18 (on page 255) the following about the geometrization conjecture:

- A non-trivial connected sum $M_1\# M_2$ admits a geometric structure if and only if $M_1=\mathbb{R}P^3=M_2$.
- "Most $3$-manifolds admit a hyperbolic structure.
- All other $3$-manifolds admit one of the other geometric structures or can be cut along $2$-spheres and $2$-tori into pices that admit geometric structures.

From this it follows that a "randomly chosen" $3$-manifold is not a connected sum.

I thought one way to make this precise is to use the hyperbolic Dehn surgery theorem:

Every $3$-manifold can be obtained by Dehn surgery along a link in $S^3$.

If the link is a hyperbolic link, then the resulting manifold is hyperbolic except for only finitly many surgery coefficients.

A knot in $S^3$ is hyperbolic if and only if it is not a torus or a satellite knot.

On Wikipedia it is written: "Every non-split, prime, alternating link that is not a torus link is hyperbolic by a result of William Menasco."

I have the following questions:

In which ways one can make claims like "a random knot is hyperbolic" precise?

Are there hyperbolic split links?

Is it true that a random link is hyperbolic? (Again how to make this precise?)

Is a random link a split link?