#OK to post
#Sam Minkin, 11/08, Assignment 17a

QUESTION #1:

(i)   2000 Presidential Election: 271 Bush - 266 Gore
(ii)  2004 Presidential Election: 286 Bush - 251 Kerry
(iii) 2008 Presidential Election: 365 Obama - 173 McCain
(iv)  2012 Presidential Election: 332 Obama - 206 Romney
(v)   2016 Presidential Election: 304 Trump - 227 Clinton

According to GFv(USEC(), x), the number of ways the election could have had the same result are:
(i)   16965465344318 ways (coeff of x^271 and x^266)
(ii)  16196017263096 ways (coeff of x^271 and x^251)
(iii) 3182416524832 ways (coeff of x^365 and x^173)
(iv)  8628577597686 ways (coeff of x^332 and x^206)
(v)   13873406885786 ways (coeff of x^304 and x^227)

QUESTION #2:

(i)   2000 Popular Vote: 47.9 Bush - 48.4 Gore
(ii)  2004 Popular Vote: 50.7 Bush - 48.3 Kerry
(iii) 2008 Popular Vote: 52.9 Obama - 45.7 McCain
(iv)  2012 Popular Vote: 51.1 Obama - 47.2 Romney
(v)   2016 Popular Vote: 46.1 Trump - 48.2 Clinton

The estimated probability of the ultimate winner according to getting 270 electoral votes is:
(i) GFvp(USEC(),479/963,x) - 0.4854668622 (add(coeff(%, x, 270 + k), k = 0 .. 268))
(ii) GFvp(USEC(),507/990,x) - 0.5464029380 (add(coeff(%, x, 270 + k), k = 0 .. 268))
(iii) GFvp(USEC(),529/986,x) - 0.6445533537 (add(coeff(%, x, 270 + k), k = 0 .. 268))
(iv) GFvp(USEC(),511/983,x) - 0.5780307623
(v) GFvp(USEC(),461/943,x) - 0.4501744769

The estimated probability of the ultimate winner according to getting the number of electoral votes they actually got is:
(i) GFvp(USEC(),479/963,x) - 271 or more - 0.4779353432 
(ii) GFvp(USEC(),507/990,x) - 286 or more - 0.4261964826
(iii) GFvp(USEC(),529/986,x) - 365 or more - 0.06752534480 
(iv) GFvp(USEC(),511/983,x) - 332 or more - 0.1599131128
(v) GFvp(USEC(),461/943,x) - 304 or more - 0.2197161242

QUESTION #3:

(i) Running using SimuCount(USEC(),p,2000,4)
(ii) Running using StatAnal(GFvp(USEC(),p,x),x,4)

For p=3/10:
(i) [163.0920000, 47.10374228, 0.1828673009, 2.749385856]
(ii) [161.4000000, 46.65683230, 0.2695069204, 2.844567676]

The values are fairly close.

For p=2/5:
(i) [214.7900000, 50.05978326, 0.04589739220, 2.641824843]
(ii) [215.2000000, 49.87825176, 0.1260503198, 2.769840597]

The third moments are somewhat different, while the others are within a small margin of error.

For p=1/2:
(i) [269.1700000, 51.05521619, 0.04519799664, 2.688441189]
(ii) [269., 50.90677755, 0., 2.748917015]

The third moments have a larger error of margin, but the rest are fairly similar.

For p=3/5:
(i) [321.0800000, 50.50504530, -0.1237192917, 2.648610096]
(ii) [322.8000000, 49.87825176, -0.1260503198, 2.769840597]

The values are fairly similar.

For p=7/5:
(i) [538., 0.]
(ii) Can't be made into a probability generating function FAIL (failed with the function)


The expected value of votes is increasing as probability of win increases, as expected. The standard deviations seem to be fairly similar no matter the probability of win. The third moments are a few magnitudes off of each other and there does not seem to be a relationship with increasing probability. Finally, the fourth moments seem to be very close to each other and stay fairly constant as probability increases.