We analyze the propensity to approve a random proposal of a large committee that makes decisions by weighted voting. The approach is a generalized version of James Coleman's "power of a collectivity to act". Throughout the paper it is assumed that the voters are of two kinds: a fixed (possibly empty) set of "major" (big) voters with fixed weights, and an ever-increasing number of "minor" (small) voters, whose total weight is also fixed, but where each individual's weight becomes negligible. As our main result, we obtain that asymptotically many minor voters act like a modification of the quota for the vote among major voters. The paper estimates the rate of convergence which turns out to be very high if the weight distribution among the small voters is not too skewed. The results obtained are illustrated by evaluating the decision rules for the Council of Ministers of the EU for various scenarios of EU enlargement. © 2007 Springer-Verlag.