Two Error Bounds of Imperfect Binary Search

Suppose we know that an object is in a sorted table and we want to determine the index of that object. To achieve this goal we could perform a binary search[1]. However, suppose it is time-consuming to determine the relative position of that object to any other objects in the table. In this scenario, we might want to resort to an incomplete solution: we could device an algorithm that quickly predicts the result of comparing two objects, and replace the actual comparison with this algorithm during a binary search. The question then is how far away are the results yielded by the imperfect binary search from the correct answers. We present two lemmas about the expected error of a imperfect binary search that goes the wrong direction with a fixed probability.