According to the official press release: “Of the 1,089 ballots cast, 691 voted against representation (“NO”) by SEIU and 398 for representation by SEIU (“YES”). There were, however, 502 ballots challenged based on issues of voter eligibility. Because the number of challenged ballots is greater than the spread between the “YES” and “NO” votes, the challenges could determine the outcome and will be subject to post-election procedures of the NLRB.”

The obvious question is what is the probability this would change the outcome of the election? If the NO’s lose 397 votes and hence the YES lose 015 on the recount the outcome will be 294 NO, 293 YES. A fraction 0.6345 of the votes were NO. We should treat this as an urn problem but to get a quick answer you can suppose the YES votes lost are Binomial(502,0.3655). In the old days I would have to trot out Stirling’s formula and compute for an hour to get the answer but now all I have to do is type into my vintage TI-83 calculator

Binompdf(502,0.3655,105) = 2.40115 X 10^{-14}

i.e., this is the probability of fewer than YES votes lost.

Regular reader of this blog will remember that I made a similar calculation to show that there was a very small probability that the 62,500 provisional ballots would change the outcome of the North Carolina election since before they were counted Cooper had a 4772 vote lead over McCrory. If we flip 62,500 coins then the standard deviation of the change in the number of votes is

{62,500(1/4^{) 1 / 2 }= 125

So McCrory would need 33,636 votes = 2386 above the mean = 19.08 standard deviations. However, as later results showed this reasoning was flawed: Cooper’s lead to a more than 10,000 votes. This is due to the fact that, as I learned later, provisional ballot have a greater tendency to be Democratic while absentee ballots tend to be Republican.

Is this all just **#fakeprobability**? Let’s turn to a court case *de Martini versus Power*. In a close electionin a small town, 2,656 people voted for candidate A compared to 2,594 who voted for candidate B, a margin of victory of 62 votes. An investigation of the election found that 136 of the people who voted in the election should not have. Since this is more than the margin of victory, should the election results be thrown out even though there was no evidence of fraud on the part of the winner’s supporters?

In my wonderful book Elementary Probability for Applications, this problem is analyzed from the urn point of view. Since I was much younger when I wrote the first version of its predecessor in 1993, I wrote a program to add up the probabilities and got 7.492 x 10 ^{-8}. That computation supported the Court of Appeals decision to overturn a lower court ruling that voided the election in this case.If you want to read the decision you can find it at

http://law.justia.com/cases/new-york/court-of-appeals/1970/27-n-y-2d-149-0.html