In which situation do we choose randomized binary search instead of the normal binary search?
Randomized binary search makes sense if your randomness source has a good bias towards your search target. You can then use it to reduce the expected search time (even retaining the $O(\log n)$ asymptotic worst-case complexity) as described in this answer.

If your randomness source is simply a uniform distribution, then you won't get much out of it. If you use it naively (i.e., simply choosing the midpoint according to your randomness source), it will actually perform worse than standard binary search since the worst-case complexity will be $O(n)$, not $O(\log n)$ (consider the case where your interval is $[1,n]$, your target 1, and your randomness source outputs $n-1, n-2, n-3, \dots$).