  Consider an election between two candidates in which the voters' choices are
random and independent and the probability of a voter choosing the first
candidate is $p>1/2$. Condorcet's Jury Theorem which he derived from the weak
law of large numbers asserts that if the number of voters tends to infinity
then the probability that the first candidate will be elected tends to one. The
notion of influence of a voter or its voting power is relevant for extensions
of the weak law of large numbers for voting rules which are more general than
simple majority. In this paper we point out two different ways to extend the
classical notions of voting power and influences to arbitrary probability
distributions. The extension relevant to us is the ``effect'' of a voter, which
is a weighted version of the correlation between the voter's vote and the
election's outcomes. We prove an extension of the weak law of large numbers to
weighted majority games when all individual effects are small and show that
this result does not apply to any voting rule which is not based on weighted
majority.
