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====== Math 445 Lab 8: Presidential election ======

Your job is to predict the outcome of today's Presidential election 
given the last-minute polling data, using Monte Carlo simulation.

Specifically, given a list of states, their electoral votes, the composite 
polling percentages for each candidate, and the margins of error those 
polling percentages, you are to run a large number of simulations of the 
election and determine the likelihood that either candidate will win based 
on the results of those simulations. For each state, start by assigning the 
specified composite polling percentages to the two candidates. Then add to
each candidate's percentage a different random number in the range between 
''-margin'' and ''+margin''. Compare the two percentages and award that state's 
electoral votes to the candidate with the larger percentage of votes. Do this 
for all fifty states (plus DC), add up all the electoral votes for each candidate, 
and award the ''n''th election to the candidate with the majority of electoral votes. 

Run a large number of such simulated elections, keeping track of the number of 
electoral votes for each candidate in each election. Make a histogram that 
shows the statistical distribution of total electoral votes for one of the 
candidates, using bins of width 10 between 0 and 540 (0-9.99 for bin 1, 10-14.99 for 
bin 2, etc). If you can figure out how, color the bins corresponding to Romney 
wins red and the bins corresponding to Obama wins blue, or else just draw a vertical 
line at the magic number of 270 electoral votes needed to win the election outright.

Then answer the following questions

  - Who is most likely to win the presidential election?
  - What is the probability that the most likely winner will actually win?
  - What is the most likely range of electoral votes for the winner? (among the bins of width 10 specified above)
  - What is the likelihood of a 269-269 electoral vote tie? 

Turn in print-outs of your codes, your histogram, and your answers to the above questions. 

Tips:

  * Start with a small number of simulated elections (say 100) and then increase to a large number (say 10,000) when you're confident your code is working correctly. 
  * You can also develop your code using simulated data, for example, just ten states all with the same polling numbers and a very small margin of error.
  * Try to use as few for-loops as possible. If you are really on fire, you can do it with just one for-loop that loops over the number of trials. 
  * Changing the colors of histogram bins in Matlab is not as easy as one might hope. You'll need to take data returned from the **hist** function and replot it with the **bar** command. See http://www.mathworks.com/matlabcentral/newsreader/view_thread/290534 for an example of how to do this.



Some further questions you might also address

  * The margins of error reported in the table are really 95% confidence levels, corresponding to two standard deviations of a Gaussian distribution. Modify your code so that the random number added to each percentage is from a Gaussian distribution with standard deviation of one-half the margin of error.  Does this significantly change your results?
  * Does doubling or halving the margins of error significantly change your results?
  * How many elections do you need to simulate in order to get reliable answers?
  * The lab as written assumes a two-party presidential election. Should we include third-party candidates? Why or why not? How would you revise your code to include a third party? Would it change the results significantly?
  * We are trusting that the polling data form an accurate estimate of the actual votes cast, to within the margins of error. The data reported below was obtained from [[http://fivethirtyeight.blogs.nytimes.com/]], and is claimed by its compiler to be unbiased and statistically reliable estimate, though there is a fair amount of controversy about this, split along party and ideological lines. Do you think the given polling data is fair and accurate? Is there a reason to suspect it is or is not?
  * Do you believe your own election prediction? Why or why not?


Relevant matlab commands; **rand**, **randn**, **sum**, **hist**, and **bar**, plus standard plotting commands such as **xlabel**, **ylabel**, **title**. 

Nate Silver, a sports statistician, pioneered the use of Monte Carlo methods 
in election prediction during the 2008 elections ([[http://fivethirtyeight.blogs.nytimes.com/]], [[http://en.wikipedia.org/wiki/FiveThirtyEight]]). In the 2008 elections, His model predicted 49 of 50 states correctly for the Presidential race (missing Indiana, which went to Obama by 1%) and all 35 Senate races correctly. Note that this lab does not cover the subtlest and most difficult aspect of election prediction: producing good composite poll numbers and margins of error from large numbers of pollsters using different methods, sample sizes, and polling dates. There is quite a bit of controversy in the current election over Mr. Silver's methods and his assessment that Obama has an 91% chance of winning the election. See, for example, [[http://cosmiclog.nbcnews.com/_news/2012/10/30/14809227-political-forecasts-stir-up-a-storm?lite]], 
[[http://www.dailykos.com/story/2012/11/01/1153661/-Nate-Silver-s-Math-Based-Math]], or google "Nate Silver controversy".


Here's some current polling data, taken from [[http://fivethirtyeight.blogs.nytimes.com]] on 2012-11-01. You can load this into Matlab as a matrix ''P'' by cutting and pasting the data into a text file ''P.asc'' and running ''load P.asc'' within Matlab. If you don't believe this polling data, feel free to use something you trust more. 
<code>
% Composite Presidential election polling numbers
% from http://fivethirtyeight.blogs.nytimes.com
% 2012-11-06 1am
%
%  O == Obama percentage 
%  R == Romney percentage
%  M == margin of error
% EV == electoral votes
%
% O    R    M    EV      state36.8  62.7  3.8   9   %  AL
38.8  59.7  6.0   3   %  AK
46.2  53.0  3.3  11   %  AZ
38.7  59.7  3.8   6   %  AR
58.2  40.5  2.9  55   %  CA
50.9  48.2  3.0   9   %  CO
56.7  42.4  3.3   7   %  CT
59.6  39.7  5.5   3   %  DE
93.1   6.3  3.2   3   %  DC  
49.9  49.7  2.7  29   %  FL
45.5  54.1  2.7  16   %  GA
66.5  32.6  3.9   4   %  HA
32.2  66.1  4.4   4   %  ID
59.9  39.5  3.0  20   %  IL
45.3  53.9  3.0  11   %  IN
51.2  47.8  3.2   6   %  IA
38.0  61.0  6.1   6   %  KA
40.4  58.7  4.5   8   %  KY
39.4  59.8  3.5   8   %  LA
56.1  42.7  3.7   4   %  ME
61.0  38.0  3.0  10   %  MD
59.1  39.8  3.7  11   %  MA
53.1  45.8  2.7  16   %  MI
53.8  45.0  2.9  10   %  MN
39.4  60.1  5.3   6   %  MS
45.6  53.6  2.8  10   %  MO
45.3  53.1  3.9   3   %  MT
40.5  58.8  3.3   5   %  NE
51.9  47.2  2.9   6   %  NV
51.5  47.8  3.4   4   %  NH
55.6  43.4  3.3  14   %  NJ
54.2  44.6  3.6   5   %  NM
62.5  36.9  2.8  29   %  NY
48.9  50.5  2.6  15   %  NC  
42.1  56.5  3.9   3   %  ND
51.4  47.6  2.7  18   %  OH
33.9  65.8  3.8   7   %  OK
53.7  44.0  3.6   7   %  OR
52.6  46.5  2.6  20   %  PA
61.9  36.3  4.3   4   %  RI
43.3  56.0  4.6   9   %  SC
42.6  56.1  4.2   3   %  SD
41.4  57.7  3.9  11   %  TN
41.3  58.1  3.1  38   %  TX
27.8  70.5  4.1   6   %  UT
66.3  32.5  4.8   3   %  VT
50.8  48.6  2.5  13   %  VA
56.2  42.5  3.5  12   %  WA
41.4  57.4  4.7   5   %  WV
52.5  46.8  2.9  10   %  WI
30.9  67.6  6.0   3   %  WY
</code>
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