A note on the probability of having a strong Condorcet winner

In an election, a strong Condorcet winner is a candidate who is top-ranked by more than 50% of the voters. The purpose of this note is to provide some algebraic representations for the probability of having a strong Condorcet winner in three-candidate elections. Three alternative procedures for gene rating voting situations are considered: the Impartial Culture condition, t he Impartial Anonymous Culture condition and the Maximal Culture condition. It turns out that the conclusions we obtain strongly depend on the way for generating voting situations.