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Comment by jvanderbot | Hacker News Reader

The question was "How likely is it that the votes worked out so well that they were basically even 1/10 percentages and not ugly numbers?"

So for a given number of votes, which determines a split, how many times does the split come out so nice? Answer: Effectively none - there are always ugly numbers with lots of decimal places.

Now that analysis comes after they conjecture that the percentages were fixed apriori. The first comment "That seems fishy" basically says this. "How can it be that we're so close to even 1/10 percentages. How can it be that we're exactly one vote off from nice 1/10 percentages"? Fishy indeed - must be rounding.

And they tell you: it's very unlikely to be 1 vote off from nice 0.1% percentage splits.

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Another way of writing it out:

How likely is it that you'd get these votes distributions

    51.2000000%
    44.2000000%
    04.6000000%

exactly? With all of those clean 0s? Very low.

But it's also possible that there was sloppy reporting and the vote counts were re-processed at some point in the chain and rounded to one decimal place.

It's more that if you start with those clean, single decimal percentages and a total number of votes, you'd end up with decimals for number of votes, which isn't possible. So if you then remove the decimal from the votes, you get slightly different percentage values when taken to 7 decimal places, but the original decimals would still be the same.

The chances of those numbers occurring normally for all 3 vote counts together is just ridiculously tiny.

The numbers are definitely sus, but what if they were

    52.2543689%
    44.2689426%
    04.6345625%

How likely is it that you'd get these votes distributions exactly with these exact tails? Compared to all other possibilities?

Basically the same 1/verymany chance, but that doesn't matter. The difference is that there's no particular reason to choose these numbers to start with. There _is_ a reason to choose nice round, but not too round numbers: that's what humans do.

The question is "How likely is it that humans would begin with those numbers when fudging?" answer is low.

But clean numbers? Much more likely.

This comment perfectly distills this post.

The same probability as any combination of three results with 7 decimals and adding up to 100.

The question isn't "probability you get those exact numbers". It's "probability that you get numbers which all have at most N decimal places".

Well there weren't zeros but within rounding error it was exact.

That actually gives a way to estimate the probability. There's 1002 choose 2 ways to divide 1000 permils over the 3 options. While there's 10 058 776 choose 2 ways to divide the 10 058 774 votes. That works out to about 1e-8 of the possible results being an exact multiple of 0.1% up to rounding error.

Of course an actual election doesn't simply pick one of the possible results at random (heck even if everyone voted randomly that wouldn't be the case). However these 'suspicious' results are distributed in a very uniform stratified fashion, any probability distribution that's much wider than 0.1% would approximately result in the same 1e-8 probability. And pretty much no reasonable person would expect a priori that the vote would result in such a suspicious number with such a high accuracy, so this should be considered strong evidence of fraud to most people.

But that’s not what happened, if you read the article it actually expands everything out to the seventh decimal, and they’re not all zeros

It is equivalent to all zeroes with the numeric precision used.

Yes, they are not all zeros, but they are exactly what you'd expect if someone picked percentages that were all zeros, then added +1/-1 to get integer votes.

So the argument is once removed, but still compelling.

Actually if you're going that many decimals its

  51.1999971%
  44.1999989%
  04.6000039%

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