If two symmetric convex bodies K and L both have nicely bounded sections, then the intersection of random rotations of K and L is also nicely bounded. For L being a subspace, this main result immediately yields the unexpected existence versus prevalence phenomenon: If K has one nicely bounded section, then most sections of K are nicely bounded. The main result represents a new connection between the local asymptotic convex geometry (study of sections of convex bodies) and the global asymptotic convex geometry (study of convex bodies as a whole). Our method relies on the recent isoperimetry of waists due to M. Gromov [G].