We study approximate ℵ 0 -categoricity of theories of beautiful pairs of randomizations, in the sense of continuous logic. This leads us to disprove a conjecture of Ben Yaacov, Berenstein and Henson, by exhibiting ℵ 0 -categorical, ℵ 0 -stable metric theories Q for which the corresponding theory Q P of beautiful pairs is not approximately ℵ 0 -categorical, i.e., has separable models that are not isomorphic even up to small perturbations of the smaller model of the pair. The theory Q of randomized infinite vector spaces over a finite field is such an example. On the positive side, we show that the theory of beautiful pairs of randomized infinite sets is approximately ℵ 0 -categorical. We also prove that a related stronger property, which holds in that case, is preserved under various natural constructions, and formulate our guesswork for the general case.