Luxembourg may have had diplomatic influence, but mathematically, Luxembourg was once powerless. This was the situation for decision-making in the voting system of the Council of Ministers of the European Economic Community of 1958-1972. The basic idea of influence or so-called voting power is the concept of decisiveness of a voter on a voting outcome. Which constellations of influence are possible? How to choose and control the distribution of voting power? How to choose voting weights and decision quota to minimize the democracy deficit in two-tier voting systems like the Council of Ministers of the European Union? The author develops and improves combinatorial and probabilistic tools to investigate voting systems, particularly two-tier voting systems with binary voting rules (“Yes” or “No”) on a theoretical level. The approaches and results contribute to designing voting systems in a more transparent way, especially with view on a fair representation and possible enlargements in the future.