 How likely is a Condorcet Paradox or any of the other voting paradoxes?  There are two ways to approach this question. The first is to calculate the probability that a majority cycle will occur in an election scenario. There is a sizable literature devoted to analytically deriving the probability of a majority cycle occurring in election scenarios of varying sizes (see Gehrlein 2006, and Regenwetter et al. 2006, for overviews of this literature). The calculations depend on assumptions about the distribution of rankings among the voters. One distribution that is typically used is the so-called impartial culture, where each ranking is possible and occurs with equal probability. For example, if there are three candidates, and it is assumed that the voters’ ballots are rankings of the candidates, then each possible ranking  can occur with probability 1/6. Under this assumption, the probability of a majority cycle occurring has been calculated (see Gehrlein 2006, for details). Riker (1982, p. 122) has a table of the relevant calculations. Two observations about this data: First, as the number of candidates and voters increases, the probability of a majority cycles increases to certainty. Second, for a fixed number of candidates, the probability of a majority cycle still increases, though not necessarily to certainty (the number of voters is the independent variable here). For example, if there are five candidates and seven voters, then the probability of a majority cycle is 21.5 percent. This probability increases to 25.1 percent as the number of voters increases to infinity (keeping the number of candidates fixed) and to 100 percent as the number of candidates increases to infinity (keeping the number of voters fixed). Prima facie, this result suggests that we should expect to see instances of the Condorcet and related paradoxes in large elections. Of course, this interpretation takes it for granted that the impartial culture is a realistic assumption. Many authors have noted that the impartial culture is a significant idealization that almost certainly does not occur in real-life elections. Tsetlin et al. (2003) go even further arguing that the impartial culture is a worst-case scenario in the sense that any deviation results in lower probabilities of a majority cycle (see Regenwetter et al. 2006, for a complete discussion of this issue, and List and Goodin  2001, Appendix 3,  for a related result). 