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Re: Electoral college probablities
Posted:
Nov 2, 2000 6:02 PM


In article <8tngpg$bsk$1@nntp.itservices.ubc.ca>, Robert Israel <israel@math.ubc.ca> wrote: [...] >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^24) = .8255161606.
Very slick, I like it. Just as a reference point for some of us who may be slightly inclined towards the concrete, here is a simulation of 20 elections with N=101, S=11, with the state by state and national totals, and electoral result, each marked + or  accordingly as the given candidate won or lost:
48 57+ 41 45 44 54+ 51+ 47 51+ 42 51+ 531 5 49 48 45 54+ 48 43 53+ 56+ 48 51+ 49 544 4 50 49 45 48 48 58+ 47 58+ 55+ 55+ 52+ 565+ 5 56+ 56+ 58+ 49 41 56+ 57+ 52+ 49 43 50 567+ 6+ 47 44 44 39 55+ 48 58+ 47 55+ 52+ 46 535 4 53+ 46 46 47 64+ 53+ 47 62+ 49 53+ 48 568+ 5 43 49 49 54+ 55+ 47 55+ 53+ 54+ 50 39 548 5 50 55+ 57+ 47 48 60+ 56+ 54+ 53+ 54+ 49 583+ 7+ 47 53+ 50 63+ 44 51+ 48 57+ 53+ 56+ 57+ 579+ 7+ 54+ 56+ 54+ 48 53+ 50 57+ 61+ 59+ 50 53+ 595+ 8+ 50 52+ 52+ 50 48 57+ 60+ 58+ 50 45 52+ 574+ 6+ 50 51+ 57+ 58+ 40 53+ 42 50 51+ 50 38 540 5 51+ 51+ 48 44 51+ 51+ 38 52+ 43 45 48 522 5 57+ 51+ 48 55+ 50 50 56+ 50 54+ 42 45 558+ 5 49 45 45 48 45 49 51+ 50 47 51+ 55+ 535 3 53+ 40 54+ 53+ 42 44 47 50 43 50 48 524 3 57+ 45 56+ 49 59+ 49 49 52+ 49 57+ 48 570+ 5 45 48 53+ 53+ 54+ 48 38 50 52+ 59+ 54+ 554 6+ 53+ 46 53+ 51+ 58+ 50 57+ 51+ 49 58+ 53+ 579+ 8+ 55+ 45 54+ 50 56+ 49 45 55+ 53+ 45 49 556+ 5
... showing 6 anomalous results.
Three runs of 10000 elections gave anomalous results 1981, 1940, and 1928 times. Your formula predicts 10000*F(0.86185771) = 1943.83 . Almost too good! ... but it was fair and square. ( Although, one might fairly argue that if they hadn't looked good, I would have tried some more. )
Lew Mammel, Jr.



