Summary: Two players are placed on a line at a distance \(d\) which is drawn from a known distribution \(F\). The players have no common notion of direction on the line, and each has a resources bound on the total distance he can travel. If \(F\) is bounded and the resources are sufficiently large, then the players can ensure a meeting. The expected time minimization problem in that case has been studied by the authors in a previous paper. Aside from that case the most the players can do is maximize the probability that they meet. This is the problem studied here, for general and specific distributions. This problem generalizes that of R. D. Foley, T. P. Hill and M. C. Spruill [Naval Res. Logist. 38, 555–565 (1991; Zbl 0732.90042)], where one of the players is stationary (zero resources).