  Consider $2k-1$ voters, each of which has a preference ranking between $n$
given alternatives. An alternative $A$ is called a Condorcet winner, if it wins
against every other alternative $B$ in majority voting (meaning that for every
other alternative $B$ there are at least $k$ voters who prefer $A$ over $B$).
The notion of Condorcet winners has been studied intensively for many decades,
yet some basic questions remain open. In this paper, we consider a model where
each voter chooses their ranking randomly according to some probability
distribution among all rankings. One may then ask about the probability to have
a Condorcet winner with these randomly chosen rankings (which, of course,
depends on $n$ and $k$, and the underlying probability distribution on the set
of rankings). In the case of the uniform probability distribution over all
rankings, which has received a lot of attention and is often referred to as the
setting of an "impartial culture", we asymptotically determine the probability
of having a Condorcet winner for a fixed number $2k-1$ of voters and $n$
alternatives with $n\to \infty$. This question has been open for around fifty
years. While some authors suggested that the impartial culture should exhibit
the lowest possible probability of having a Condorcet winner, in fact the
probability can be much smaller for other distributions. We determine, for all
values of $n$ and $k$, the smallest possible probability of having a Condorcet
winner (and give an example of a probability distribution over all rankings
which achieves this minimum possible probability).
