All rights reserved by the author; a permission to copy the whole article or any part of it is granted for non-for-profit purposes, provided the title, the author's name, and this copyright note are included in the copy.

The fact that Florida has been the last, chronologically, "undecided" state in presidential race 2000 produced a false impression that the Sunshine State is the only uncertain element of the election process that can shake its final outcome. This certainly is not the case, and in order to find a scientifically sound solution of the current controversy one needs to look at five other close calls as well. These are: Iowa, New Hampshire, New Mexico, Oregon, and Wisconsin. Restricting considerations to Florida only, while neglecting the other five close races, will inevitably lead to statistically invalid conclusions that can only be accepted by those unfamiliar with probability theory. (Even a brief observation of cash flow in any gambling casino may prompt one to believe that many of those playing are indeed unfamiliar with elementary probability theory, for otherwise they would be more hesitant to bet in a game they are statistically certain to lose, and this is exactly what the casino operators are hoping for; perhaps, this also is the reason why has Florida monopolized the media's attention.)

President Bill Clinton said: "The American people have spoken, but we aren't quite sure what they said". I can add to this observation that, most likely, we never will. And the best we can do in the current situation is to mathematically evaluate statistical chances that either of the three possible outcomes of the election (the third one being a stalemate) did actually take place. In this paper, we demonstrate that Gov. George W. Bush actually won the election 2000 with probability at least 6.4% greater than the probability that Vice President Al Gore won it. This margin is about a raw of magnitude higher than in Florida, where the difference in these probabilities is not greater than 1.6%.

Measurements, as long as the degree of complexity of the measured matter is large enough, have to be associated with an error. This is particularly true in the area of data acquisition where both the human factor and the mechanical factor are sources of unavoidable mistakes. Humans are not always objective, in that they sometimes see what they would like or expect to see, as opposed to what there really is, and some will have a tendency to doctor the raw data so that they better conform to the theses they are supposed to support. Moreover, tiredness and distraction is a common source of many unintentional human mistakes. On the other hand, mechanical devices, particularly those with moving parts or those extracting data from paper-based media (e.g., keyboards, punched card readers, and scanners) tend to have a much higher error rate than the ones associated with digital electronic computers (which aren't completely error-free, either). Because of these reasons, 100% accurate acquisition of a huge amount of data, although theoretically possible, is hardly ever practical or economically feasible. Fortunately, this fact doesn't mean that such an error-prone acquisition process must be necessarily inconclusive. To the contrary, contemporary mathematics provide us with useful tools that allow for astonishingly close approximations of inaccurately measured physical quantities.

A measurement error seemed to take place in counting of the votes in this year's presidential election (at least the election's results have been contested based on a claim that the error existed and was substantial). It must have been obvious to any bystander with sufficient mathematical background that having the exact count of about 100 million votes cast, on paper ballots, for all the candidates is infeasible, at least economically. And although attention of arguing parties, as well as mass media, has been focused almost exclusively on the results of the Florida presidential race, the question is more complex than just who won how many votes in Palm Beach and Broward counties (which, in itself, is far from being simple a problem).

It has been announced that nationally, roughly 2.1 million votes, or 2.1%, have been rejected in the data acquisition phase, which calls any outcome of state election closer than that 2.1% in question. We may certainly believe that any error, should it occur, was distributed evenly among all groups of voters, but from probability theoretic standpoint, it's highly unlikely that it really was. Since the recounting in these close races does not seem to be a rational solution (costly, legally dubious, and very error-prone to the extent that the new results may be even farther from the actual numbers that the original ones), can we do something to draw a firm conclusion from these uncertainties? The answer is "Yes". This paper presents a probability theoretic analysis of the election results that leads to firm conclusion that the winner of the presidential election 2000 is Gov. George W. Bush. All the details of the calculations, including a program computing the probabilities (in the Appendix), are elementary enough to be verified by a college student. They are included at the end of this article, so that those who disagree may easily tell me when and where I went wrong (if I did). The row data are readily available from the internet.

There were six races where the differences between the winner and the loser were less than 2.1%, which I assume in this paper to be the upper bound of an error, although there were suggestions the actual average error was roughly 1%. If the difference were greater than 2.1% then it would be extremely unlikely that the error could have changed the result, since even if it totally unfairly "favored" one candidate against the other, it could have accounted for at most 2.1% change - not enough to swing the outcome. Should the difference be less than 2.1% then, with a positive probability, the result could have been falsified by the error, and the smaller the difference the higher the probability that it actually had been.

The six questionable races are:

Florida, 25 electoral votes
Number of Gore votes: 2910942
Number of Bush votes: 2911872
Bush leads by 0.02%

Iowa, 7 electoral votes
Number of Gore votes: 638355
Number of Bush votes: 634225
Gore leads by 0.32%

New Hampshire, 4 electoral votes
Number of Gore votes: 265853
Number of Bush votes: 273135
Bush leads by 1.35%

New Mexico, 5 electoral votes
Number of Gore votes: 286112
Number of Bush votes: 285933
Gore leads by 0.03%

Oregon, 7 electoral votes
Number of Gore votes: 719165
Number of Bush votes: 712705
Gore leads by 0.45%

Wisconsin, 11 electoral votes
Number of Gore votes: 1240431
Number of Bush votes: 1235035
Gore leads by 0.22%

Because of the aforementioned errors, the actual number of votes and the winners in each of these races may have been, with a non-neglectable probability, different then indicated by these figures, with Florida and New Mexico being relatively most likely candidates for such discrepancies. (The next least difference, beyond these six, has been recorded in Minnesota where Gore has a lead of 2.54%, and even though the figures recorded there, 1168190 for Gore and 1110290 for Bush, are, most likely, inaccurate, he assumed 2.1% error is very unlikely to affect the outcome of the race. Other states have differences between 3.42% and 76%, so that the outcomes must be assumed statistically certain there as well.) In particular, it is highly doubtful that the actual number of valid votes that were genuinely cast in Florida for any of the two contestants is identical with the original or recounted results, and it seems obvious that further recounts, just like playing the same old vinyl disk over and over again, will permanently alter the subject matter (the ballots, that is) and will only increase the discrepancy between the record and the reality.

Seemingly the easiest and the most natural approach would be to prorate the questionable 59 electoral votes accordingly to the totals of votes cast in the six states for each candidates. One could hope that any relative bias would be better neutralized in a larger sample. Such an expectation is indeed consistent with the so called "large number" hypotheses widely accepted by the statisticians. Straightforward calculations (a program that carries them out is included in the Appendix) produce a surprising, although not unexpected result: since the total vote (5,928,160 for Bush, 5,988,617 for Gore) in these states splits 49.7/50.3, both Bush and Gore get roughly half of 59, or to be exact, 29.4 and 29.6 (no cruelty intended) electors, respectively. Because Bush has accumulated more certain votes than Gore did, this pronounces Bush's victory: Bush gets total 271.4 versus Gore's 266.6.

The problem with prorating is that is seems to go against the idea of electoral vote as opposed to popular vote. Let's see what can we do to better follow the constitutional framework designed by Founding Fathers.

Because, except for Bush's 1.35% lead in New Hampshire, all differences are less that one half of a percent, or are about one fifth or less of the margin of error, let's just try to toss a coin for each of these races and see what we get. There are 64 possible outcomes of such a game, each of them having a probability of 1/64. Some of them will make Gore winner; the sum of their probabilities is the Gore's chance that he won the election. Some of them will make Bush winner; the sum of their probabilities will be the Bush's chance. Two of them will result in a stalemate (32 for Gore and 27 for Bush); their probability of 1/32, or 3.1% is a chance that the election 2000 was actually a tie.

The program in the Appendix computes all these numbers. The results are:

• Probability that Bush won: 53.1%

• Probability that Gore won: 43.8%

• Probability that it was a tie: 3.1%

So, Bush has a higher chances than Gore by a margin of more than 9% - much above the margin of error - thus giving him the advantage of preponderance of evidence.

The coin flipping method may not fly well with some skeptics. So, let's rerun our program for refined probabilities for Gore and Bush in the six states. Let's assume that the random variable Z with mean value m and standard deviation sigma adequately models the statistical error of data acquisition during the election process. Following classic assertion from mathematical statistics, experimentally validated by countless test cases, let's assume that Z has a normal distribution of probability (a.k.a. the bell curve). The normalcy assumption means that the error has no bias, and that the "correct" value is distant from the measured mean m no more than by error sigma with probability more than 84%. In each race, mean m is the recorded percentage-point difference between Bush and Gore, and deviation sigma is the assumed error of 2.1%. Moreover, let's notice that, conspiracy theories aside, the probabilities for different races are mutually independent (in that who won in New Hampshire does not depend on who won in Wisconsin, etc.) so that to obtain the probability of a combination of outcomes one must simply multiply the particular probabilities. Using these assumptions and any statistical software package (e.g., http://ebook.stat.ucla.edu/calculators/cdf/normal/normalcalc.phtml), we obtain:

Probability that Bush actually won Florida: 50.4% (so Gore's chances 100% - 50.4% = 49.6%).

Probability that Bush actually won Iowa: 43.9% (which leaves 56.1% for Gore).

Probability that Bush actually won New Hampshire: 74.0% (which leaves 26.0% for Gore).

Probability that Bush actually won New Mexico: 49.4% (which leaves 50.6% for Gore).

Probability that Bush actually won Oregon: 41.5% (which leaves 58.5% for Gore).

Probability that Bush actually won Wisconsin: 45.8% (which leaves 54.2% for Gore).

Note that in any particular state the probability of a stalemate is practically 0, while the probability of a stalemate in the Electoral College is not.

Running the same program (see the Appendix) with these refined probabilities yields the following results:

Probability that Bush won = 52.8%

Probability that Gore won = 43.1%

Probability that it was a tie = 4.1%

So, again, Bush has a higher chances than Gore by a margin of more than 9% - much above the margin of error.

For comparison, below are the results for error of 3.1% (for those who may believe that the 2.1% rejection rate could, to its all unlikeliness, add to a "rule of thumb" error of 1%), 1% (a "rule of thumb" error of punched cards voting hardware), 0.5% (a statutory threshold for automatic recount in some states), 0.33% (the minimum difference between the probabilities of Bush's and Gore's wins), 0.1% and 0.01% (to see the trend). The first case (3.1%) includes, additionally, the State of Minnesota because the winner's lead in that state (2.54%) was less than 3.1%.

It turns out that in entire interval, the probability of Bush's win is never less than 52.3% (being equal to when the error is 0.5%) while the probability of Gore's win is never more than 46.3% (when the error is 0.3%), with the minimum difference between the two roughly equal to 6.4% (when the error is 0.33%). In all cases, Bush's lead is about an order of magnitude bigger than the margin of error.

sigma = 0.01
Probability that Bush won = 97.7%
Probability that Gore won = 2.3%
Probability that it was a tie = 0.0%

sigma = 0.1
Probability that Bush won = 57.9%
Probability that Gore won = 42.1%
Probability that it was a tie = 0.0%

sigma = 0.33
Probability that Bush won = 52.6%
Probability that Gore won = 46.2%
Probability that it was a tie = 1.2%

sigma = 0.5
Probability that Bush won = 52.3%
Probability that Gore won = 45.0%
Probability that it was a tie = 2.6%

sigma = 1.0
Probability that Bush won = 52.6%
Probability that Gore won = 43.3%
Probability that it was a tie = 4.1%

sigma = 2.1
Probability that Bush won = 52.8%
Probability that Gore won = 43.1%
Probability that it was a tie = 4.1%

sigma = 3.1
Probability that Bush won = 57.1%
Probability that Gore won = 39.8%
Probability that it was a tie = 3.1%

FURTHER IMPROVEMENTS

It is possible, for instance, based on statistical random tests of the voting equipment used in the six states, to further refine the values of probabilities that Bush won (Gore's chances are equal to one minus the Bush's chances) in these states. Also, the standard error might have been different in several states, even in several counties, and the error distribution might have not been normal, for instance, it might have been biased (which almost certainly would have been a case if manual recounts or a massive voter fraud took place). One can take these new refined probabilities and plug them into the program from the Appendix, thus obtaining the modified results. Taking into account the level of technological sophistication of the equipment used, wide elimination of human factor from the process, as well as the openness and public scrutiny of the process of voting and counting, it seems rather unlikely that these refined probabilities would depart considerably from ours. Therefore, it's our belief that because of the two digit percentage point margin obtained with the current probabilities, the result would have been the same, that is, that Bush actually won the election with probability of more tan 50% plus the margin of error.

APPENDIX

COMPUTATIONS

Here is the actual output of the program that shows how the main results were computed. The numbers in first six lines were imputed. The program text can be found in the next section.

Probability Bush won in FL:.504
Probability Bush won in IO:.439
Probability Bush won in NH:.74
Probability Bush won in NM:.494
Probability Bush won in OR:.415
Probability Bush won in WI:.458

Bush has 242 and needs 28 additional electoral votes

Binary sequences represent possible outcomes in 6 states

1 = win, 0 = loss for Bush

000000 end of sequence
Prob = 0.0116071 Bush Electors = 0

100000 end of sequence
Prob = 0.0117943 Bush Electors = 25

010000 end of sequence
Prob = 0.0090829 Bush Electors = 7

110000 end of sequence
Prob = 0.0092294 Bush Electors = 32

001000 end of sequence
Prob = 0.0330356 Bush Electors = 4

101000 end of sequence
Prob = 0.0335684 Bush Electors = 29

011000 end of sequence
Prob = 0.0258514 Bush Electors = 11

111000 end of sequence
Prob = 0.0262683 Bush Electors = 36

000100 end of sequence
Prob = 0.0113318 Bush Electors = 5

100100 end of sequence
Prob = 0.0115146 Bush Electors = 30

010100 end of sequence
Prob = 0.0088675 Bush Electors = 12

110100 end of sequence
Prob = 0.0090105 Bush Electors = 37

001100 end of sequence
Prob = 0.0322521 Bush Electors = 9

101100 end of sequence
Prob = 0.0327723 Bush Electors = 34

011100 end of sequence
Prob = 0.0252383 Bush Electors = 16

111100 end of sequence
Prob = 0.0256454 Bush Electors = 41

000010 end of sequence
Prob = 0.0082341 Bush Electors = 7

100010 end of sequence
Prob = 0.0083669 Bush Electors = 32

010010 end of sequence
Prob = 0.0064434 Bush Electors = 14

110010 end of sequence
Prob = 0.0065474 Bush Electors = 39

001010 end of sequence
Prob = 0.0234355 Bush Electors = 11

101010 end of sequence
Prob = 0.0238135 Bush Electors = 36

011010 end of sequence
Prob = 0.0183390 Bush Electors = 18

111010 end of sequence
Prob = 0.0186348 Bush Electors = 43

000110 end of sequence
Prob = 0.0080388 Bush Electors = 12

100110 end of sequence
Prob = 0.0081685 Bush Electors = 37

010110 end of sequence
Prob = 0.0062906 Bush Electors = 19

110110 end of sequence
Prob = 0.0063921 Bush Electors = 44

001110 end of sequence
Prob = 0.0228797 Bush Electors = 16

101110 end of sequence
Prob = 0.0232487 Bush Electors = 41

011110 end of sequence
Prob = 0.0179041 Bush Electors = 23

111110 end of sequence
Prob = 0.0181929 Bush Electors = 48

000001 end of sequence
Prob = 0.0098082 Bush Electors = 11

100001 end of sequence
Prob = 0.0099664 Bush Electors = 36

010001 end of sequence
Prob = 0.0076752 Bush Electors = 18

110001 end of sequence
Prob = 0.0077990 Bush Electors = 43

001001 end of sequence
Prob = 0.0279157 Bush Electors = 15

101001 end of sequence
Prob = 0.0283659 Bush Electors = 40

011001 end of sequence
Prob = 0.0218449 Bush Electors = 22

111001 end of sequence
Prob = 0.0221972 Bush Electors = 47

000101 end of sequence
Prob = 0.0095756 Bush Electors = 16

100101 end of sequence
Prob = 0.0097300 Bush Electors = 41

010101 end of sequence
Prob = 0.0074932 Bush Electors = 23

110101 end of sequence
Prob = 0.0076141 Bush Electors = 48

001101 end of sequence
Prob = 0.0272536 Bush Electors = 20

101101 end of sequence
Prob = 0.0276932 Bush Electors = 45

011101 end of sequence
Prob = 0.0213268 Bush Electors = 27

111101 end of sequence
Prob = 0.0216708 Bush Electors = 52

000011 end of sequence
Prob = 0.0069580 Bush Electors = 18

100011 end of sequence
Prob = 0.0070702 Bush Electors = 43

010011 end of sequence
Prob = 0.0054448 Bush Electors = 25

110011 end of sequence
Prob = 0.0055326 Bush Electors = 50

001011 end of sequence
Prob = 0.0198034 Bush Electors = 22

101011 end of sequence
Prob = 0.0201228 Bush Electors = 47

011011 end of sequence
Prob = 0.0154968 Bush Electors = 29

111011 end of sequence
Prob = 0.0157468 Bush Electors = 54

000111 end of sequence
Prob = 0.0067929 Bush Electors = 23

100111 end of sequence
Prob = 0.0069025 Bush Electors = 48

010111 end of sequence
Prob = 0.0053157 Bush Electors = 30

110111 end of sequence
Prob = 0.0054014 Bush Electors = 55

001111 end of sequence
Prob = 0.0193338 Bush Electors = 27

101111 end of sequence
Prob = 0.0196456 Bush Electors = 52

011111 end of sequence
Prob = 0.0151293 Bush Electors = 34

111111 end of sequence
Prob = 0.0153733 Bush Electors = 59

Probability that Bush won = 52.8%

Probability that Gore won = 43.1%

Probability that it was a tie = 4.1%

PROGRAM elections(INPUT,OUTPUT);

by

Marek A. Suchenek

All rights reserved by the author; a permission to copy the whole program or any part of it is granted for non-for-profit purposes, provided the author's name, and this copyright note are included in the copy.}

var ElectVotes, Sequence: ARRAY [1..7] of INTEGER;
I, BushNeeds, GoreNeeds: INTEGER;
State: ARRAY [1..7,1..2] of CHAR;
CountBush, CountGore, ProbEven: REAL;
ProbBushCurrent: ARRAY [1..7] OF REAL;

PROCEDURE AddVotes (remaining: INTEGER; Prob: REAL; Sum:
INTEGER); BEGIN {AddVotes}
IF remaining = 0 THEN
BEGIN {THEN}
FOR I := 1 TO 6 DO
write (Sequence[I]:1);
writeln (' end of sequence');
writeln ('Prob = ', Prob:10:7, ' Bush Electors = ', Sum:1); IF
Sum >= BushNeeds THEN CountBush := CountBush + Prob ELSE
IF Sum <= BushNeeds - 2 THEN CountGore := CountGore + Prob;
END {THEN}

ELSE
BEGIN {ELSE}
{case 1: Gore wins, Bush gets no electoral votes}
Sequence[remaining] := 0;
AddVotes (remaining - 1, Prob*(1.0 -
ProbBushCurrent[remaining]), Sum); {end of case 1} {case 2:
Bush wins and gets electoral votes} Sequence[remaining] := 1;
AddVotes (remaining - 1, Prob*ProbBushCurrent[remaining],
Sum + ElectVotes[remaining]);
END; {ELSE}
END; {AddVotes}

BEGIN {elections}
ElectVotes[1] := 25;
ElectVotes[2] := 7;
ElectVotes[3] := 4;
ElectVotes[4] := 5;
ElectVotes[5] := 7;
ElectVotes[6] := 11;
State[1,1] := 'F';
State[1,2] := 'L';
State[2,1] := 'I';
State[2,2] := 'O';
State[3,1] := 'N';
State[3,2] := 'H';
State[4,1] := 'N';
State[4,2] := 'M';
State[5,1] := 'O';
State[5,2] := 'R';
State[6,1] := 'W';
State[6,2] := 'I';

END. {elections}