Suppose there are N candidates instead of three. What is the probability of a runoff?

Question

On Nov. 3, the residents of X City will elect a mayor from among three candidates. The winner will be the candidate who receives an outright majority (i.e., more than 50 percent of the vote). But if no one achieves this outright majority, there will be a runoff election among the top two candidates.

If the voting shares of each candidate are uniformly distributed between 0 percent and 100 percent (subject to the constraint that they add up to 100 percent, of course), then what is the probability of a runoff?

Suppose there are N candidates instead of three. What is the probability of a runoff?

Answer 1

I analyzed the problem in the following manner.

Let X1, X2, XN be U[0,1], then let the percentage of Xi be Xi/∑Xi

All variables less than Xmax will be U[0, Xmax]

Then, to figure the probability that the sum of all Xi < Xmax be greater than Xmax, note that we can transform the variables, and let Xmax = 1, Xi ~ U[0, 1}, Xi not equal to Xmax

Then, the volume of ∑Xi <= 1 = 1/(n-1)!, and the volume over all the Xi is 1, so the probability that there will be a runoff is 1 - 1/(n-1)! or (n-1! - 1)/n-1!