Elementary probability models can sometimes be used to analyze the results of an election where irregularities have occurred. In this article we give an illustrative example by considering the 1975 mayoral election in the city of Flint, Michigan. True vote is esti- mated with an estimator which is used in randomized response models.
KEY WORDS: Election challenge; True vote estimation; Random- ized response model.

The result of a close election may be challenged on the basis of irregularities and the outcome may ultimately be decided by the courts. A given court may uphold the result, reverse the result, invalidate the entire election, or offer any of a number of remedies. (For a comprehensive review of election law including postelection remedies, see "Developments in the Law-Elections" (1975).) In general, an election will not be overturned on the basis of a mere mathe- matical possibility that the results would be reversed in the absence of irregularities. Being reluctant to unjustly disenfranchise the valid electors, the courts have sometimes required that a challenger establish a probability that the result would be reversed in the absence of irregularities.
Sometimes the evidence consists of proof that a certain number of persons voted who were not quali- fied, with no evidence as to how these persons voted. Finkelstein and Robbins (1973) compute the probability of reversal under random removal of a number of votes equal to the number of illegal votes. They point out that this is a neutral and proper measure of probability of reversal in cases where there is no evidence to indicate that one candidate or another benefited from the illegal votes.
In this article we give the facts associated with the contested mayoral election in the city of Flint, Michi- gan, on November 4, 1975. The irregularity in this election is not a common one; however, the problem it created provided a situation where an elementary probability model could be used to quantify the probability of reversal.

The city of Flint, Michigan, held an election on November 4, 1975 for the office of Mayor, wherein a mayor was to be elected for the first time under a new city charter. The vote for mayor after recount is shown in Table 1. The breakdown for Precincts 51 and 52 is given because these vote totals were disputed due to a mixup in ballot assemblies in these precincts. The vote totals from the other 143 precincts and absentee voters were not disputed. The recorded vote shows Rutherford the winner with a margin of 206 votes. However, due to the errors in ballot assemblies in the two precincts and the closeness of the vote, the decision by the Board of Canvassers to declare James Rutherford the winner over Floyd McCree was chal- lenged in the courts.
Those who voted in person used punch-card devices in the following manner. The individual voter was given a punch-card to insert in a ballot assembly. The ballot assembly is part of the voting device and is a book with the names of candidates printed on the pages with a hole beside the name of each candidate. The in- dividual voted for the candidate of his choice by punching through the hole with a stylus. This re- moved a square from the punch card. The computer used to tabulate the vote was programmed to know the correspondence between squares on the punch card and votes for specific candidates.
The ballot assemblies were arranged among pre- cincts so that neither candidate would always be listed first or second. The computer was programmed and the individual cards coordinated to take the rotation into account. Precincts 51 and 52 had five and four voting devices, respectively. The ballot assemblies were to be rotated between the two precincts so that the candidate whose name was first in Precinct 51 was second in Precinct 52.
By mistake, the election officials placed one ballot assembly in Precinct 51 that should have been in Precinct 52, and one ballot assembly in Precinct 52 that should have been in Precinct 51. The result was that a voter using the device with the wrong assembly in either precinct would have his vote recorded for the other candidate.
The error in ballot assemblies was discovered after the polls had closed and the voting devices were dis- assembled. The commingling of all punch-cards from voting devices within a precinct made it impossible to distinguish which or how many votes were improperly recorded (reversed) within Precincts 51 and 52.