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b1shop comments on Probability and Politics - Less Wrong
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I don't like the coin model because it ignores replacement.
Assume there's ten other people in a room. Six like red and four like blue. Four of them will go to the polls, and you're trying to decide if you should, too. What's the probability your vote will be the deciding factor?
It's tempting to use the binomial distribution. p=0.5, n=4. Your vote matters if x=2.
So it'll be tied without you about 35% of the time.
But this is incorrect. If the first person who votes casts a red ballot, then the probability the next vote is red falls to 5/9, and the probability the next vote is blue increases to 4/9. The correct model is the Hypergeometric model because it doesn't assume replacement.
It computes a higher 43%.
As n increases from 10 to 300000000, I imagine the effect is more dramatic.
Let me refocus on my point. I want to estimate the probability my vote will matter.
With population n, participation rate v, and pre-election polling showing r support for the policy, the probability your vote will matter is equal to:
(C[nv/2,nr]C[nv/2,n(1-r)])/C[n,nv]