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Re: Electoral college probablities
Posted:
Oct 31, 2000 5:23 PM
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In article <G38p38.439@cwi.nl>,
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 israel@math.ubc.ca
Department of Mathematics http://www.math.ubc.ca/~israel
University of British Columbia
Vancouver, BC, Canada V6T 1Z2
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