Letx1,x2, …xn, … be a sequence of independent random variables with a common density functionP(x= 1) =p,P(x= 0) = 1 –p, 0 <p< 1. This paper considers the non-randomized sequential procedures δ's for estimatingpand the following three problems on choice of δ. (i) Choose δ to minimizeEv(Zδ–P)2subject toEpNδ≤mwherem≥ 1 andZδis an unbiased estimate ofp; (ii) Choose δ to minimizeEpNδsubject toEp(Zδ–p)2≤ α where α is a real positive number; (iii) choose δ to minimizeCEpNδ+Ep(Zδ–p)2whereCis the cost of an observation. In each case the minimization is to be done uniformly inpif possible; otherwise the supremum overpof the risk in question is to be minimized. A procedure is constructed for problem (i) whenmis not an integer. A fixed sample size procedure is shown to be admissible and minimax for problem (ii). A procedure is constructed which is asymptotically uniformly better than the fixed sample size for problem (ii). Furthermore, for problem (iii) some optimum procedures are constructed.