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Comment by bbctol | Hacker News Reader

I'd at least want some idea of what to compare it to. That's certainly closer than if we assigned percentages at random. On the other hand, my intuition is they're less close than if every voter was picking at random, though I'm not sure if either of those counterworlds is relevant to the argument.

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Each voter picking at random will get you almost exactly 50/50 every time, because the sample size is so large. The expected variance goes with the square root of the sample size, so with 100 million voters, you'd expect the outcome of each voter choosing randomly to be about 10,000 off from exactly 50/50.

Here's a quick Python snippet to test it empirically:

    print sum(random.randint(0, 1) for x in xrange(0, 100000000))

(I strongly recommend using pypy to run it. CPython takes forever to complete.)

With 8 runs, I got these results:

All of which require at least two decimal places in the percentage to be distinguishable from 50% each way.

Pretty amazing. If the outcome is determined by 100 million independent coin flips, the probability of a margin greater than one tenth of a percent is 1.5e-23. In R

    n      <- 100000000
    diff   <- 0.1/100
    a_wins <- n*(1-diff)/2
    b_wins <- n*(1+diff)/2
    p1 <- pbinom(a_wins, size = n, prob = 0.5)
    p2 <- pbinom(b_wins, size = n, prob = 0.5, lower.tail = F)
    p1 + p2 # = 1.523959e-23

Parent is saying if we randomly chose the outcome, not the votes.

Sure. But why should we expect random percentages to have any relationship to reality?

(I would argue that random votes isn't related to reality either...)

Huh? "than if every voter was picking at random" Sounds like what I showed to me.

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