The various vote-counting systems follow the same general
method, as diagrammed below. A voter "marks" his
input, by punching a hole, coloring a bubble, or touching a
circle on a screen; the system reads that input, by human or
computerized means; & then the system updates its tally,
again by human or computerized means.
voter ---> input ---> system ---> tally
Each of the arrows is a potential source of error. The voter
could mark the wrong bubble or leave a hanging chad. A human
system could misread a ballot, or a computerized system could
be calibrated wrongly. Finally, a human system could put a
mark under the wrong candidate's name, or a computerized
system could add to the wrong name, subtract, or run out of
storage space. It should be noted that as regards
computerized systems, any errors almost surely have human
roots.
If we concentrate on the latter two types of error, those
which would be the fault of the system or its designers, we
should expect that a fair system would not favor or disfavor
any given candidate, or mathematically, that the probability
of an error favoring, say, Bush equals that of an error
favoring Kerry. Or, if we are given that we have an error of
this type & that it favors either Bush or Kerry, the
probability of it favoring Bush, P(B), should equal the
probability of it favoring Kerry, P(K). We would expect P(B)
= P(K) = 1/2.
So if we could come up with an unbiased list of system errors
occuring in this past election, errors that favor either Bush
or Kerry, we could then test whether those errors concur with
the "fairness hypothesis," i.e. P(B)=P(K)=1/2.
Note that this problem is exactly analogous to the problem of
determining whether a coin is fairly balanced. If it is, then
P(heads) = P(tails), & we would not expect to see,
intuitively, ninety heads out of a hundred throws, while sixty
out of a hundred would not be too surprising. A more formal
approach uses mathematics & probability.
Suppose we have n cases of this type of error. Of those,
denote by b the number that favor Bush. The probability that
we would see b or more errors in favor of Bush is given by the
binomial distribution:
P(X >= b) = SUM[nCk/(2^n), k=b,...,n].
Some sample numbers:
n=10, b=8 ==> P(X >= 8)=0.0547
n=15, b=12 ==> P(X >=12)=0.0176
n=20, b=15 ==> P(X >=15)=0.0207
As you can see, it is pretty unlikely to see fifteen heads
flop out of twenty throws; it would likewise be pretty
unlikely for fifteen of twenty fair errors to favor Bush.
The tough part of the problem is determining what n & b
are. There have been several cases reported here on DU, &
most of them indeed favored Bush. But I am not convinced this
is a representative sample of all such errors. If anyone here
knows of an unbiased, respectable website that has compiled
not only system errors that turned in Bush's favor but also
those that helped Kerry, then we could plug-in some numbers
for n & b, & determine to within a fair degree whether
or not the heretofore detected errors suggest a systematic
unfairness in the errors of our voting systems.
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