A "random" isometry, like taking a sheet of paper from a table and randomly laying it back, "almost surely" is a rotation or a glide reflection (they have three degrees of freedom). This applies regardless of the details of the probability distribution, as long as θ and the direction of the added vector are independent and uniformly distributed and the length of the added vector has a continuous distribution. A pure translation and a pure reflection are special cases with only two degrees of freedom, while the identity is even more special, with no degrees of freedom.