Powers of subgroups in voting bodies

This paper discusses the power p _n of an n-member subgroup B _n of an N-member voting body, N odd and 1 n N. In contrast to bloc voting, we assume that the members vote independently with equal probability for and against a given issue. Power p _n is defined as the probability that the outcome of a vote changes if all members of B _n reverse their votes. Theorems: p _{n + 1} = _n for odd n < N; p _n + p _{N – n} = 1; P _m + p _n > p _{m + n} if m + n < N; p _{n + 1}/p _n (n + 1)/n as N for fixed even n; for rational 0 > > 1, p _N 2^–1 sin^{–1 1/2} as N . A simple summation formula is given for p _n .