In article <G38p38.firstname.lastname@example.org>, Peter L. Montgomery <Peter-Lawrence.Montgomery@cwi.nl> wrote: > In the US presidential election, the winner is determined >by electoral vote rather than by popular vote. >What is the chance that the two will give different outcomes >in an otherwise close race?
> Make simplifying assumptions, such as an odd number >of equi-populous states. >In a two-candidate race, voters everywhere vote randomly >for one of the two candidates. >Every state has the same (odd) number of voters, >and the same number of electors -- >the winning candidate within each state gets all of that state's electors. >What is the probability that the candidate winning >a majority of the overall vote will lose in a majority of the states?
An interesting problem. Let's say there are S states, each with N voters, and S is fairly large. Let X_i be the (signed) difference between candidate 1 and candidate 2 in state #i, and E_i the event that candidate 1 wins state #i. Let M be the signed difference in popular vote, D the signed difference in states, and W the event that candidate 1 wins more states.
Then E[X_i | E_i] = m = (N+1) (N choose (N+1)/2)/2^N. Of course E[X_i | not E_i] = -m, and E[X_i^2 | E_i] = E[X_i^2 | not E_i] = N.
E[M | D] = ((D+N)/2) m + ((N-D)/2) (-m) = D m. E[D | W] = s = (S+1) (S choose (S+1)/2)/2^S (the same formula as for m), and E[D^2 | W] = E[D^2 | not W] = E[D^2] = S. Thus E[M | W] = m s E[M^2 | W] = E[M^2 | not W] = E[M^2] = SN
Assuming S is large, we can probably use a normal approximation, so given W, M is approximately normal with mean m s and variance SN - (ms)^2. So the probability that the winning candidate loses in the popular vote is approximately F(-m s/sqrt(SN-(ms)^2)), where F is the standard normal cdf.
For example, with S=51 and N=10^6 we would have s = 5.726033806 and m = 797.8847608, and F(-m s/sqrt(SN-(ms)^2)) = F(-.8323716341) = .2025996041.
Asymptotically, m = sqrt(2 N/Pi) and s = sqrt(2 S/Pi), so -m s/sqrt(SN - (ms)^2) -> -2/sqrt(Pi^2-4) = -.8255161606.
Robert Israel email@example.com Department of Mathematics http://www.math.ubc.ca/~israel University of British Columbia Vancouver, BC, Canada V6T 1Z2