A point is placed at random on the real line according to some known distribution F, and a search is made for this point, beginning at some starting points s on the line, and moving along the line according to some function x(t). The objective of this article is to maximize the probability of finding the point while traveling at most d units. Characterizations of simple optimal searches are found for arbitrary distributions, for continuous distributions with continuous density everywhere (e.g., normal, Cauchy, triangular), and for continuous distributions with density which is continuous on its support (e.g., exponential, uniform). These optimal searches are also shown to be optimal for maximization of the expected number of points found if the points are placed on the line independently from a known distribution F.
