<p>We study fractional distributionally robust probability maximization (DRPM) problems that maximizes the worst-case probability that the ratio of two stochastic functions exceeds a fixed threshold. We explore a general phi-divergence based ambiguity set in which the worst-case distribution has an uncertain distributional support. The problem is formulated as a semi-infinite optimization problem including a distributionally robust adaptive chance constraint in which the probability level to be attained is a decision variable. We propose a comprehensive reformulation method and show that the DRPM problem can be recast as a biconvex mixed-integer nonlinear programming (MINLP) problem for any phi-divergence ambiguity set when the ratio has a concave-to-convex form. Furthermore, we show that the boundedness of the perspective function of the phi-divergence conjugate function is a sufficient condition under which we can reformulate the biconvex problem into an equivalent finite-dimensional convex MINLP problem. We consider four commonly used phi-divergences: the Kullback-Leibler divergence, Chi-square divergence, Total Variation distance, and Hellinger distance. For the Kullback-Leibler and Chi-square divergences, the boundedness of the perspective function is necessary to derive a convex MINLP reformulation. In contrast, for the Total Variation and Hellinger divergences, the boundedness requirement is not needed to derive a convex MINLP reformulation.</p>