Suppose that all states have an 80% probability of voting for Obama.
If they are independent, the odds of an Obama win are .8 * 8 * .8 + 3 * .8 * .8 * .2=.896
However, suppose the states are maximally DEPENDENT-- they always vote the same way. Then, the odds of an Obama win are exactly .8
The effect will be much stronger with more states, hence the gigantic difference between XKCD's result, and intrade's odds on an obama vs mccain win. In the limit of an infinite number of states, 51% odds in each + independence would give Obama a 100% chance of winning!
Things are much more complex, but this is basically the point cperciva was making. It is almost certainly true that the states are not independent. For example, the odds in Ohio are factoring in the chance of some gigantic gaffe or a terrorist strike or whatever, that would change the vote everywhere.
Suppose that all states have an 80% probability of voting for Obama.
If they are independent, the odds of an Obama win are .8 * 8 * .8 + 3 * .8 * .8 * .2=.896
However, suppose the states are maximally DEPENDENT-- they always vote the same way. Then, the odds of an Obama win are exactly .8
The effect will be much stronger with more states, hence the gigantic difference between XKCD's result, and intrade's odds on an obama vs mccain win. In the limit of an infinite number of states, 51% odds in each + independence would give Obama a 100% chance of winning!
Things are much more complex, but this is basically the point cperciva was making. It is almost certainly true that the states are not independent. For example, the odds in Ohio are factoring in the chance of some gigantic gaffe or a terrorist strike or whatever, that would change the vote everywhere.