In the world of real algebraic geometry there are natural probabilistic questions one can ask: you can make sense of a random hypersurface of degree d in some projective space and ask about its expected topology where "expected" makes sense because there are sensible measures on the space of hypersurfaces. See Welschinger-Gayet http://arxiv.org/abs/1107.2288 and http://arxiv.org/abs/1005.3228 for recent progress on such questions (e.g. what is the expected Betti number of a random real hypersurface of degree d?).

In geometry more generally you might want to make statements like "a general manifold is aspherical" or "a general manifold has positive simplicial volume". It seems difficult to construct sensible measures for which these questions have answers: to talk about probability you need some way of producing manifolds (and then distinguishing them) in a random way.

However, Cheeger proved that for fixed L there is only a finite set $D_L$ of diffeomorphism classes of manifold admitting a Riemannian metric with sectional curvatures bounded above in norm by L, volume bounded below by 1/L and the diameter bounded above by L (see the first theorem in Peters "Cheeger's finiteness theorem for diffeomorphism classes of Riemannian manifolds" http://www.reference-global.com/doi/abs/10.1515/crll.1984.349.77). This means that you can ask questions like "what is the average total Betti number of a manifold in $D_L$" and "how does this increase with L?" (are there exponential upper bounds?), or one can try to make sense of the limit as $L\rightarrow\infty$ of the proportion of manifolds in $D_L$ with zero simplicial volume.

Are there any known concrete answers to these questions, or other formulations of the questions which lead to answers?