59 divisions in Philly with no votes for Romney a statistical possibility?

A viral meme had stated that no votes for Romney in 59 divisions in Philadelphia is a mathematical and statistical impossibility. PolitiFact, a publication of Poynter Institute, had analyzed the claim and ruled it a "Pants on Fire."

PolitiFact based its judgment on small numbers of registered Republicans in those divisions:

Some of the 59 divisions had fewer than 10 registered Republicans, according to records from April 2012. The average number of registered Republicans in these divisions was 17.

The number 17 is indeed not big, not hundreds or thousands. But 59×17=1003. What is the probability that all 1003 registered Republicans in 59 divisions in question did not vote? According to Pennsylvania Department of State the percent of registered voters voting in 2012 elections was 67.63%. Thus the probability for 1003 voters not voting is 0.3237¹⁰⁰³≅5×10^-492. In other words it is zero.

Above I have computed the probability that all of the Republicans in given 59 divisions did not vote. However it is proper to compute the probabilty that none of the Republicans in any 59 out of 1,700 Philadelphia divisions did not vote. For this purpose we need to compute the binomial coefficient with n=1,700 and k=59. In the case n≫k we can approximate it as n^k/k!. After substituting the numbers we get for the binomial coefficient ≅3×10¹¹⁰.

We obtain the probabilty that none of the Republicans in any 59 out of 1,700 Philadelphia divisions did not vote by multypling the probability that all of the Republicans in given 59 divisions did not vote by the binomial coefficient and get ≅10^-381. Again a zero. Note that in this calculation I implicitly assumed that all 1,700 Philadelphia divisions have about 17 registered Republicans just like those 59 with zero Republican vote. Since this is an obvious underestimate our final number is even an overestimate.

My conclusion is that PolitiFact's "Pants on Fire" ruling was unfounded.

Mikhail Simkin
August 22, 2022

Related:

A list of precincts with over 99% vote for Obama in 2008 elections.

A comment on “No evidence for systematic voter fraud: A guide to statistical claims about the 2020 election” and “Statistical detection of election irregularities”