In this paper, we address a sampling procedure for approximating the Deegan-Packel index, which is particularly advantageous when tackling with large-scale problems. It is based on the estimation of an expectation in a subdomain. Its performance is analyzed in terms of statistical properties and of bounds for the incurred absolute error. The effectiveness of this tool is demonstrated through a real-world example where this power index can be exactly computed. Finally, we use it to determine a new power distribution for two compositions of the Board of Governors of the International Monetary Fund (IMF). The resulting distributions of the power are compared with the ones prescribed by the Shapley and the Banzhaf values.