How were ancient leaders chosen in rome?

they supposedly flipped a coin and who ever was able to tell you the correct answer, either heads or tails, was the winner of their tournament or election.

Another way they did it was people would randomly put up their hands as wanting to run in the election and, at random, the senators would pick the one with either the most amount of confidence or just at random.

(or whoever appealed the most to them)
ancient Rome covered a lot of time and they had three different forms of government over their long history, During the first phase or the monarchy, the kings were chosen by heredity or force of arms (which ever worked). During the republic the officials were elected by the citizens in their voting assemblies. During the principate, the emperor would appoint leaders and, ideally, name a heir to the throne. Towards the end of the western empire, military force worked to establish who was emperor and how long he stayed one.