LIKELIHOOD OF ALTERING THE OUTCOME OF THE FLORIDA 2000
PRESIDENTIAL ELECTION BY RECOUNTING
We seek to estimate the probability that a manual recount of one
or several counties, or of the entire state, will produce a change
in the outcome of the election to choose Electors from Florida for
the election of a US President in 2000. Data are current as of
Monday Nov 13 2000.
The thrust of the argument is that a manual recount of votes in Palm
Beach County can be expected to add votes favoring the candidate
already leading there (Gore); justice might then demand a similar
action in all counties, presumably adding more votes for each
candidate in proportion to their current vote counts statewide (about
49% each). Since a manual recount is expensive and time-consuming, it
is necessary to ask whether or not there is a reasonable probability
that this process will produce a change in the net lead of one
candidate over the other of a magnitude sufficient to change the
outcome of the contest. We estimate the odds against this happening to
be about two-to-one. The actual odds are subject to discussion; the
general conclusion (i.e. that there is a nontrivial possibility of
reversal) is nonetheless clear.
Naturally the political decision about whether or not to proceed with
such a recount is then a subjective decision: given these odds that
the effort would change the choice of President of the United States,
is it appropriate to invest the necessary time and money in a
statewide manual recount? We do not attempt to answer this question,
but note that today's news includes announcements requiring all
recounts to be completed by Nov 14 at 5pm, obviously precluding manual
recounts statewide.
Please note that the analysis below is expected to match the situation in
Palm Beach county well. We extend our analysis to the statewide numbers,
but there are significant county-by-county differences we have not yet
incorporated; these are likely to weaken the validity of our claims about
statewide vote counts. (Our analysis is ongoing.)
Disclaimer: I am a mathematician, not a lawyer, nor a resident of Florida.
BACKGROUND
The latest Palm Beach machine-tabulation shows candidate Gore receiving
269,732 votes, and candidate Bush 152,951. The current statewide totals
are Gore 2,909,907, Bush 2,910,195. Given no other information,
our best guess would be that these are the correct numbers of votes
actually cast (that is, intended and actuated) by the voters.[^1] We know,
however, that machine tabulation introduces a number of systematic
errors -- instances in which the tabulated vote count differs from the
numbers of votes cast.[^2] We wish to consider the likelihood that these errors
are large.
We know there is a fraction of votes which are not recorded
correctly. The hand count of 4695 votes on November 11 2000 found
49 more votes when assessed by sight than were counted by machine. While
there are many sources of error in vote tabulation, the one most often
mentioned in this race is the incomplete removal of chad; in this case the
type of enumeration error will rather uniformly be the non-tabulation of an
otherwise acceptable ballot.[^3] We use this sample to estimate that there
is an error rate of approximately 1% in the vote count.[^4] It is important
to observe that this phenomenon is most likely a systematic error, just
as likely to affect votes in favor of either candidate.[^5] Therefore, it
is in most elections not a significant concern: the net effect on an
evenly-divided race is likely to be nil. However, the vote in this
particular contest is very close and thus it is appropriate to measure
just _how_ likely is this "likelihood to be nil".
ILLUSTRATION
Let us first illustrate with an idealized situation which is easier to
visualize. Suppose that there were 500,000 votes cast for the two
candidates, and that the votes cast were evenly split between the two.
One could imagine placing the votes onto a balance, at each moment
showing the net amount by which Bush (say) is favored over Gore. The
initial reading is zero; the final reading would also be zero in this
illustration as the vote is a tie. Indeed, the intermediate vote counts
are more likely to show a difference of zero than any other single difference.
If indeed the vote is evenly split, it is even possible to arrange the
voters deliberately so that the votes are even balanced periodically
(say, after every 100 votes).
Now let us consider the effect of random error: assume about one vote
out of each hundred -- with equal probability from each side -- will
fail to be enumerated. In this case, after each 100 votes, we will NOT
show an even balance of votes; rather, one side or the other -- affecting
each side randomly -- will lose a vote and the net lead will shift
one more vote towards the other side. (In a very small number of sets
of 100 votes there can be two or more errors, which will not noticeably
affect our analysis.)
We thus obtain a model in which every set of 100 votes can be viewed
as a step randomly taken in either direction on the number line. The
full set of 500,000 voters amounts to 5000 steps along the line.
It is possible that all 5000 steps would be taken in one direction,
resulting in an apparent lead of 5000 votes for one side, despite that
fact that the true numbers of votes for the two sides are equal. However,
this is vanishingly unlikely. It is also possible that the 5000 steps
would be exactly evenly split into errors favoring one side and errors
favoring the other. This would result in a vote count which accurately
reflects the even balance between the voters. However, it too is
quite unlikely: only in about one such election out of ninety would the
errors cancel each other exactly.
The odds are just slightly less in favor of a net gain of 2, 4, 6, ...
for either party. In particular, the odds are about 50-50 that this
process will result in a net lead of about 90 or less for one party;
equally well, the odds are about even that one candidate will show a
lead of about 90 or more. We might say that the initial data establish
a 90-vote threshold: a hand recount is as likely as not to correct
a 90-vote difference in this model.
Now, there are other ways to model this situation; they produce somewhat
different numerical conclusions. The key point is that it is highly probable
that an evenly-split electorate of this size can, with unbiased tabulation
errors, produce an apparent lead of several dozens of votes; and it is
highly improbable that these errors will produce an apparent lead
of several hundreds of votes.
ANALYSIS OF PALM BEACH COUNTY
In a more general analysis, we may assume the number of votes for the
two candidates to be G for Gore and B for Bush, and assume the
error rate to be e; then the net lead of Gore (say) will be G-B,
before the effect of the errors, which will number about (e)x(B+G),
and which we assume will harm the candidates in the same proportion
as their votes. Thus the measured lead will be reduced by about a factor
of 1-e. However, we can compute as well the range of values in
which a change is likely or not, as with the 90-vote threshold in the model.
Working backwards from the given data, and assuming as before a (rather
speculative) 1% rate of lost votes, we expect there were about 426,952 valid
votes in Palm Beach County for the two main candidates -- a recovery of
4269 votes -- and we expect those to favor Gore in the same ratio as
in the machine tabulation (1.7635 to 1), that is, 1545 votes for
Bush and 2724 for Gore, a net gain of 1179 votes for Gore.
This is mere extrapolation; what we wish to point out is that we
compute the probability of a gain of exactly 1179 to be only about 0.62%;
a more accurate representation of the situation is that the Gore
gain should be expected to be, say, between 1135 and 1223 votes.
Please note that this is an approximate statistical analysis; the
logic is sound but the numerical estimates are hardly better than
order-of-magnitude since we do not have detailed information about the
nature of the tabulation errors.
ANALYSIS OF STATE OF FLORIDA
The Bush campaign has pointed out that there is the potential for
abuse of the system if only Democrat-dominated counties are allowed to
recover lost votes with a manual recount. This may be true. Certainly
if all counties used the same punch-card system of balloting, we
should expect the same loss rate in all counties. As it happens, there
is a mixture of voting systems used, not all with the same failure
rate. Let us ignore this issue for now. If we assume a statewide 1%
loss rate, affecting the parties equally, then we would conclude that
some 60,000 votes could be recovered with a manual recount, roughly
equally distributed between the two major parties. Thus, at first blush,
we would _expect_ to see Bush's lead widen by just two or three votes.
However, what is significant in this analysis is that if indeed there
were very nearly 3 million votes cast statewide for each of Bush and
Gore, then a random process which erases 1% of the votes of each
party equally is as likely to show a lead of 165 or more for one
candidate or the other as it is to keep the lead correct to less than
165 votes. We calculate the odds that the lead be reported at 250 or more
to be about two to one; the odds that the lead would be reported to be
300 or more are about four to one.
In other words, there is a good chance that the Bush lead statewide
is more or less an accident resulting from a random process which by
chance produced slightly more errors among the Gore votes than among
the Bush votes, although the particular estimate used here for the
error rate suggests somewhat less than even odds that this would happen.
Clearly if the Bush lead were to widen by several hundred more votes
as a result of a tally of the overseas absentee ballots, the chance
that this would represent a true Gore lead overlaid with a random
distribution of errors is greatly diminished. (The likelihood that
these assumptions would by chance report an 800-point lead when in
fact the Gore vote exceeds the Bush vote is on the order of several
hundred to one.)
What is the likely effect of the use of different balloting systems
statewide? Among the systems likely to produce a smaller number of
lost-vote errors are the optical scanning systems similar to those
used for multiple-choice school exams, for example. If we assume these
systems produce no error likely to be caught in a hand recount, we
must adjust our analyses as follows: each candidate's total vote count
will consist of two parts -- the votes in "error-free" counties and
the votes in "error-prone" counties. If each candidate has a nearly
equal total vote count in error-free counties (and thus a nearly equal
total among error-prone counties, too) our preceding analysis applies:
the effect of a manual recount will primarily be to enlarge the total
number of votes of each, and we must accept a certain range of vote
spreads as being a likely consequence of random vote loss in the
error-prone counties. (Mathematically, the size of this spread is
roughly proportional to the square root of the number of votes cast
in those counties.)
However, if one candidate's strength lies primarily in error-prone
counties and the other's in error-free counties, the effect of a
statewide recount would be to boost the vote differential in favor of
the former. To illustrate this, simply consider the effect of a statewide
recount in which Palm Beach county is affected as in the previous
section, and the other 66 counties are assumed to be error-free:
the change in the vote totals would be exactly as discussed above.
[This analysis was written without the benefit of a breakdown of
the counties according to the type of voting and vote-tabulating systems
used. Since then I have been directed to the State's site,
election.dos.state.fl.us
which makes that information available. I will incorporate this
information into my analysis as soon as possible. A quick glance
suggests that indeed the use of punched cards is positively correlated
with the level of support for Gore. This will _definitely_ imply
that a statewide recount would tend to favor Gore (assuming, as usual,
the simple model we have built for error-correction by hand count). --djr]
CONCLUSIONS
Let us reiterate that the numerical results obtained are sensitive
to several key assumptions. We have assumed here that a manual recount
will correct an undercounting which affects all votes with equal
probability, and that this probability is .01. Small variations in
these assumptions will surely change the conclusions quantitatively,
but the qualitative conclusions will just as surely prevail:
(1) A manual recount of votes in Palm Beach county only
will likely shift the lead by perhaps a thousand or more
votes towards the Gore column
(2) A manual recount of all votes statewide will likely produce
a smaller shift of the lead
(3) There is a distinct though minority possibility that the correct
vote count statewide actually favors Gore and that the tabulated
vote showing a Bush lead is the result of a random tendency of
the tabulation errors to eliminate Gore votes.
(4) It is highly unlikely that a manual statewide recount would
reverse a lead as large as has been projected after the
overseas ballots are counted.
(5) The likelihood of a reversal of the vote increases with the
differential of support in punched-card versus error-free counties.
[This is likely to substantially alter the odds of a reversal.]
We leave the determination of the correct course of action to the
political and legal establishment.
Prof. David Rusin
Director of Undergraduate Studies
Department of Mathematical Sciences
Northern Illinois University
DeKalb, Illinois, 60115 USA
Email rusin@math.niu.edu
Web math.niu.edu
Telephone 1-815-753-6739
Fax 1-815-753-1112
==== Footnotes ==========
[^1] For the purposes of this discussion, we assume that a ballot clearly
marked for Buchanan, for example, was intended to be a vote for Buchanan.
It has been alleged that this was frequently untrue, but we are not
able to assess these allegations with this analysis.
[^2] The difference is assessed by comparing the machine count of ballots
to a visual inspection of the ballots. We are in this document assuming
that the result deduced from visual inspection is the vote intended by
the voter. This is somewhat problematic as the human interpretation of
validity of a vote is subjective. It has been alleged that this form of
error is at least as common as machine-tabulation error. For the purposes
of this document we shall assume that there is a mechanism to determine
the true intended vote; we are assessing here whether or not to undertake
that determination based on the spread of the votes.
[^3] This source of error tends to diminish on a second count as bits
of chad are loosened and freed; the effect would be similar but lessened
on a third count. Exactly these tendencies have usually been observed in
the recounts, some multiple, done in all the Florida counties.
[^4] For comparison we mention the lower 0.5% rate which triggers an
automatic recount and the higher 2%-5% quoted by elections professionals
as commonly accepted for unbiased errors in non-controversial elections.
[^5] One may allege that errors of these or other types of miscounts
tend to strike votes for one candidate preferentially. It is difficult
to assess these claims from the limited data available, but certainly
a differential error rate will affect the true vote totals to be
inferred from the measured data.
Links to further technical analyses of the Florida vote count may
be found at
madison.hss.cmu.edu |
