In this talk, the focus is on optimizing strategies for the efficient computation of the inner loop of the classical double-loop Monte Carlo for Bayesian optimal experimental design. We propose the use of the Laplace approximation as an effective means of importance sampling, leading to a substantial reduction in computational work. This approach also efficiently mitigates the risk of numerical underflow. Optimal values for the method parameters are derived, where the average computational cost is minimized subject to a desired error tolerance. We demonstrate the computational efficiency of our method, as well as for a more recent approach that approximates using the Laplace method the return value of the inner loop. Finally, we present a set of numerical examples showing the efficiency of our method. The first example is a scalar problem that is linear in the uncertain parameter. The second example is a nonlinear scalar problem. The last example deals with sensor placements in electrical impedance tomography to recover the fiber orientation in laminate composites.