Tuesday night after the 4.7 million votes had been counted from all 2704 precincts Roy Cooper had a 4772 vote lead over Pat McCrory. Since there could be as many as 62,500 absentee and provisional ballots, it was decided to wait until these were counted to declare a winner. The question addressed here is: What is the probability that the votes will change the outcome?

The do the calculation we need to make an assumption: the addition votes are similar to the overall population so they are like flipping coins. In order to change the outcome of the election Cooper would have to get fewer than 31,250 – (4772)/2 = 28,864 votes. The standard deviation of the number of heads in 62,500 coin flips is (62,250 x ¼) ^{1 / 2} = 125, so this represents 19.09 standard deviations below the mean.

One could use be brave and use the normal approximation. However, all this semester while I have been teaching Math 230 (Elementary Probability) people have been asking why do this when we can just use our calculator?

Binomcdf(40000, 0.5, 28864) = 1.436 x 10^{-81}

In contrast if we use the normal approximation with the tail bound (which I found impossible to type using equation editor) we get 1.533 x 10^{-81}.

We can’t take this number too seriously since the probability our assumption is wrong is larger than that but it suggests that we will likely have a new governor and House Bill 2 will soon be repealed.