In this paper we study the possibility of constructing satisfactory social choice mechanisms whose outcomes are determined by a combination of voting and chance. The following theorem is obtained: if a social choice mechanism does not leave "too much" to chance and satisfies a unanimity condition, then it is either uniformly manipulable or dictatorial. The result contributes to the program suggested by Gibbard [2] for the study of the extent to which social choice mechanisms in which chance plays a role can be freed from strategic manipulation.