Two measuring devices are available to classify units into one of two mutually exclusive categories. The first device is an expensive procedure which classifies units correctly; the second device is a cheaper procedure which tends to misclassify units. To estimatep, the proportion of units which belong to one of the categories, a double sampling scheme is presented. At the first stage, a sample ofNunits is taken and the fallible classifications are obtained; at the second stage a subsample ofnunits is drawn from the main sample and the true classifications are obtained. The maximum likelihood estimate ofpis derived along with its asymptotic variance. Optimum values ofnandNwhich minimize the measurement costs for a fixed variance of estimation and which minimize the precision for fixed cost are derived. This double sampling scheme is compared to the binomial sampling scheme in which only true measurements are obtained.