Suppose we wish to estimate the mean of some polynomial function of random variables from two independent Bernoulli populations, the parameters of which. rhemselves, are modeled as independent beta random variables. It. is assumed that the t.otal sample size for the experiment is fixed, but that the number of experimental units observed from each population may be random. This problem arises, ior example, when estimating the fault tolerance of a system by testing its compomentc individually. Using a decision theorebic approach, we seek to minimize the Bayes risk that arises from using a squared error loss function The Bayes estimator can lw detrmined in a straightforwardmanner, so the problem of optimal estimation rcduces. therefore, to a problem of optimallocatton of the samples between the two populatiorls. This can be solved via dynamic programming. Similar programming techniques are utilized to evaluate properties of a number of ad hoc allocation strategies that might also be collsidered for use in this problem.Two sample polynomials are analyzed along with a number of examples indicating the effects of different prior parameter settings. The effects of differences between prior pararueters used in the design and analysis stages of the experiment are also examined. For the polynomials considered, the adaptive strategies are found to be especially robust. We discuss computational techniques that facilitate such analyses by permitting rapid re-evaluation of strategies. Capabilities of this sort enrouragepeople to explorr designs more fully and to consider them from a number of different viewpuillts.