The conventional group testing problem is that of correctly classifying each member of a given population as defective or non-defective. A conventional binary group test is a simultaneous test on a subset of the population with only two possible outcomes. A ''good'' reading indicates that all the members of the subset are non-defective, and a ''bad'' reading shows that there is at least one defective member in the subset. The goal is to design an efficient algorithm to correctly identify all the defective members of a population. In this paper, we introduce the idea of generalized binary binomial group testing. The generalized group tests provide different information about the number of defectives in a group than does the conventional group test. In particular, motivated by problems in finite-user random-access communication systems, we investigate the following two generalized binary group tests: the so-called conflict/no conflict test which indicates whether there is at most one defective item in a group, and the so-called success/failure test which indicates if there exists exactly one defective item in a group. We introduce and analyze group testing procedures for the above generalized group testing problems. The proposed procedures perform better than the scheme of testing each item individually and the algorithms based on binary tree search methods. Optimality of the proposed algorithms is also discussed.