Why not just count the votes?

As you probably heard, Conservative candidate Wai Young lost on election night by 33 votes. After a partial judicial recount, she is now down by 22 votes. This whole story seems ridiculous to me. I first heard of this from Blue Like You, where Joanne seems to be covering this whole story really closely.

What strikes me as really stupid is that the Liberal victory margin narrowed by 11 votes after 28 of 184 ballot boxes were recounted. So if you assume that these ballot boxes were of average size to the rest of the ballot boxes (given no information, this seems fair), the margin of 22 votes could easily be made up in the remaining 156 boxes. There seemed to be 0.39 vote shift/ballot box, which implies there would 61 additional votes assuming the same distribution (or 72 total votes shift, which is greater than the original margin of 33). Now, there obviously is a chance the 11 vote gain of the Conservatives after the first 28 ballot boxes was an anomaly, but it isn't so hard to believe that 22 more votes could be made up in the remaining 85% of the ballots. That is why these must be counted.

What could possibly be the point of doing a random sampling of the ballot boxes that shows the results may be reversed and then stopping? If after counting 28 ballot boxes, it was shown that there was a zero vote shift, at least there would be some logic in stopping.

As an example, when an auto part manufacturer does a random sampling of their product and they find there is an unacceptable variance in the size of their parts, they don't go, "Ok, well that's good to know, but we should just keep going with what they are doing." No, they will either assume there was systematic error in the process or they will further measure to see if the original sample was representative of the whole. The point of doing a random sampling to see if there is error. Here there is shown to have been sufficient error in the first 15% of ballot boxes that would imply the remaining 85% should be counted.

Enough bloggers have covered all the stuff about every vote should count, democracy at risk, yada yada yada , but the math of it just annoys me!


Nerdy Update:
So there were 4 changes made in 22 of the 28 ballot boxes, which means there is on average 3.14 (mmm pi) vote changes per ballot box (in whatever direction). So what is the chance that the Conservative candidate can make up 22 more votes in the remaining 156 ballot boxes?

Well there is expected to be 490 vote changes in the 156 ballot boxes using the same distribution as the first 28 ballot boxes. By the Central Limit Theorem, the mean of n independent and identically distributed variables converge to a normal distribution (trust me, I wish I had to look this up). Assuming the expected change is zero overall and 50% chance each ballot can go Conservative, the Z score from the binomial distribution is:

Z = (22-0) / ((50%(1-50%)*490)^1/2) = 1.987

So this puts the chance of a 22 vote swing to the Conservatives still within the 99% confidence interval.


Nerdy Update 2:
It just occurred to me that it would be useful to see if the chance of a 11 vote swing out of 88 changed ballots seems reasonable. Let's test the hypothesis that the 11 vote swing could happen with zero being the expected vote swing.

Z = (11-0) / ((50%(1-50%)*88)^1/2) = 2.345

Which is less than 1%, or outside the 99% confidence interval. We can safely reject the null hypothesis that there is a zero expected vote swing.

This is not to say there was an issue with the initial vote counting outside of random variability since rejection of the null hypothesis does not mean that the opposite is true, but it certainly doesn't give confidence that that remaining 85% of ballot boxes need not be counted.


Very Late Update 3:
I saw some more details later and basically half of my assumptions were all incorrect. Many of the changed votes were from the previously spoiled ballots and it was 4 total ballot changed from 22 boxes (what about the other 6 boxes that were opened?). So what the new analysis would show is that the probability of making up 22 votes in the rest of the boxes is probably fairly low. However, it is still stupid to just do a partial recount.