Classfication of randomized algorithms
An example of such an algorithm is randomized Quick Sort, where you randomly permute the list or randomly pick the pivot value, then use Quick Sort as normal. Quick Sort has a worst case running time of $O(n^{2})$, but on a random list has an expected running time of $O(n\log n)$, so it always terminates after $O(n^{2})$ steps, but we can expect the randomized instance to terminate after $O(n\log n)$ steps, always with a correct answer.
This gives an subset of Las Vegas algorithms. Las Vegas algorithms also allow for the possibility that (with low probability) it may not terminate at all - not just terminate with a little bit more time.
These in turn are really just a type of Monte Carlo algorithm, where the answer can be incorrect (with low probability), which is at least conceptually different to maybe not answering.


There's a whole bunch of detail I've left out of course, you might want to look up the complexity classes ZPP, RP and BPP, which formalise these ideas.