The binomial group-testing problem is extended to the case in which the common probabilitypof a unit being defective is unknown. A Bayes “non-mixing” procedureR(1)is derived and compared with other procedures, in particular with the corresponding procedureR1that requires the knowledge ofpand with another procedure based on continually revising the maximum likelihood estimate ofp; the latter is called an empirical Bayes solution. Finally, a Bayes procedure derived in the Appendix allows “mixing” and this is conjectured to be the unrestricted Bayes solution; the improvements due to “mixing” are shown to be small in Table III. Several applications of the general problem of group-testing are discussed in the introduction and this virtually constitutes a general review of the subject. After the procedureR(1)is defined a detailed illustration is given in section 2.1 showing how to carry out this procedure with the use of Tables I and II. Lower bounds for the Bayes risk associated with any group testing procedure for unknownpare given in Table III; one of these bounds, based on information theory, is derived in section 4.