Statistical bias and the probability of an outcome. A town referendum has occurred. The question posed to voters was YES or NO on a local law. There were 3 methods of voting: Electronic machine (voting booths), absentee ballot, and affidavit ballot.
The results are as follows:
A) The machine vote tallied 13,891 “YES” vs. 13,526 “NO” votes.
 B) The absentee ballot vote tallied 377 “YES” vs. 201 “NO” votes.
 C) The affidavit ballot vote tallied 419 “YES” vs. 1,854 “NO” votes.
Questions:
1) What is the probability, given the results of the machine and absentee votes, of the affidavit ballot results occurring randomly? 
2) What is the probability of getting anywhere from 1,854 "NO" votes to 2,273 "NO" votes randomly?
Note: this is a real life application. More info regarding the scenario can be made available upon request.
 A: Historically, across time and geographical regions, votes by
machine or paper ballot (on election day) and votes by absentee
ballot have shown significant differences in percentages for
candidates, propositions, parties, etc. I think there is less
data about the voting patterns of affidavit voters. However different demographic
groups vote by different methods, and they may have different opinions. So it would not be astonishing
to see significant differences in your town. In this context
I would be cautious using the term 'bias' without  careful definition of what I meant by it.
You are asking whether two categorical variables are 'associated'
rather than 'independent': Outcome (Yes vs NO) and method of
voting (Machine, Absentee, Affidavit). A traditional way to
check for independence is to do a chi-squared analysis of a
contingency table. Here is the result of such an analysis
using Minitab statistical software.
 Chi-Square Test: Mach, Abs, Aff 

 Expected counts are printed below observed counts
 Chi-Square contributions are printed below expected counts

           Mach       Abs       Aff       Total
     Y     13891      377       410       14678 
           13299.41   280.38    1098.22
           26.316     33.299    431.285

     N     13526      201       1854      15581
           14117.59   297.62    1165.78
           24.791     31.369    406.289

 Total     27417      578       2264      30259

 Chi-Sq = 953.349, DF = 2, P-Value = 0.000

Here is an interpretation of one cell of the table: Yes votes
from Affidavit voters. Based on the total votes for various
categories, one would expect about 1098 Yes votes, compared
with only 410 actually seen. The 'contributions' in each
cell are a way to measure the importance of the discrepancy
between expected and observed votes in a cell. This is the
cell with the largest such contribution. The P-value < 0.0005
indicates a very small probability of seeing this overall
pattern of votes if the people in each category of voting
had the same Yes/No split in opinion.
However, we have so much data here (even in the Absentee and Affidavit groups) that we would be able
to detect even very small differences in preferences among the
people in different groups.
In particular, it seem quite clear the the pattern of Yes/No
votes among Affidavit voters is different from the pattern
in the other two categories. I don't know exactly what you
mean when you ask if they voted 'at random'. Presumably, they
voted according to their beliefs rather than by tossing coins.
If there is a question whether Affidavit voters had a 'right'
to vote their opinions, that is a legal issue, not a statistical issue.
I don't know what you mean by 'random' in question (b).
If one were to toss a fair coin 2264 times, then the
probability of getting between 1858 and 2273 Heads is
less than $10^{-16}$, but I don't see how that relates to
the voting question.
