Just to make very, very explicit a basic error the OP made, Nate Silver’s 90.9% odds was not that every single state would go the way he said it would. It was the odds that enough electoral votes total would go Obama’s way.
There are a lot of ways the latter could have happened.
So trying to compound probabilities for each state going one particular way does give a very small chance. But that particular scenario is only one of many, many ways the totals could have added up for an Obama win.
There were a lot of scenarios where Obama could win, a lot where Romney could win. The Obama scenarios totaled up in statistical terms to be substantially more likely.
One of the real tricks that Silver and other similar predictors used is not having to calculate the odds for every single scenario and then add them up. There are far too many scenarios to do each individual calculation. (Think of a scale of 2[sup]51[/sup] starting cases.) So various tricks are used to collapse the computational tree, each person having their own bag of tricks.
On the page MikeS linked to, there’s a graph titled Tipping Point States, which is “The probability that a state provides the decisive electoral vote.”
Is that just defined by what time the Networks call the state, with the first state called that put the victor over 270 (when added to all the states already called)? So it’s partly a function of how quickly a state counts its votes. Or is there some more technical definition?
(As an example of a different definition, multiply Obama’s votes by (1-x) and multiply Romney’s by (1+x), with x slowly increasing from 0, and see which state changes the election outcome when it flips.)
No it has to do with what would happen if the race really were close, and one state decided the election:
In this election, Colorado, ended up being the tipping point. If Romney had done better across the board, and won Virgina, Ohio, and Florida, then it would’ve been Colorado that won Obama the election.
Although you’re right as far as it goes, I’m not sure you’re drawing the salient distinction. Two random variables are either independent or not, yes, but they’re also either uncorrelated or not — the problem seems less that “independence is absolute” but rather that we lack one preferred objective measure of dependence.
At the risk of complicating the discussion, I want to note that’s only true in particular circumstances. E.g. if x ~ Uniform(0, 1) then Cov(x, x^2) = 1/12 != 0. Of course you’re right that in general uncorrelated does not imply independent.
Thanks. I searched for “Tipping Point” on that page, but didn’t see any link to a definition. That does look to be equivalent to the definition I gave in my last paragraph.