Why the map and median electoral vote (EV) calculations do not match

The question I am asked most often is why the map sometimes does not match the median electoral-vote calculation. This occurs when state races are toss-ups, and happens for the following reason.

I do a calculation that is fundamentally different than what you see in other analyses. Other analysts decide who is ahead in each state. Then they assign that state's electoral votes to the winning candidate, one state at a time. I also do this, but I do it for every possible outcome, and calculate how likely that outcome is. Then I list all possible outcomes in order of EV count (and weighted by likelihood), and see what count is in the middle of the list - the median.

This takes into account the fact that in a close race, polling data are not accurate enough to tell us with total confidence who will win, and we would be making a mistake to assign each state with certainty. To understand why this is, consider the following simplified example.

Imagine that in a race between candidates K and B, K needs to win at least one of Florida or Ohio to win the presidency, and other states do not matter. (This is not that far from the truth, and is simplified to make a clear example.) If K's chance of winning either one is 40% (odds 3-to-2 against), then his probability of winning at least one is higher, 64% (odds 16-to-9 in favor). In this case his median number of electoral votes would be more than 270 EV, but a map would show both Florida and Ohio red. In fact, the sum of electoral votes in states with probability greater than 50-50 for K would be less than 270 EV.

The numbers I give in the box are the median of all possible outcomes. This means that if polling were accurate and unbiased, the median would be the best estimate of the final outcome. Because the map shows the probability on a state-by-state basis, it shows the single most likely exact possibility. However, since so many possibilities exist, even this combination of states is not that likely. As of Friday, October 29 the probability of the precise outcome in the map is only 8%, or about 11-to-1 against.

This is an example of compound probability. We run across compound probability every day. An example like the Florida-Ohio one happens when we shop for eggs at the store. Individual eggs are usually not broken, but in a whole carton it is quite common to find a broken egg. So you should always look inside the carton before you buy.

An example in which compound probability works against you is the lottery. Picking two or three winning numbers is not that hard, but picking six is nearly impossible. In this case several events must all happen at once, which is the converse of the examples given previously.

Probability and statistics are to me a subject of great fascination because they use math to represent the real world, with all its lumps and imperfections. Several of you have asked me for names of introductory books to this subject. I don't have a suggestion yet, but if you have one please send it to me. Suggested by several readers: The Cartoon Guide to Statistics by Gonick and Smith.

For meta-analysis, try Lipsey & Wilson's Practical Meta-analysis. One reader, Blair T. Johnson, has written this: work.

Bonus explanation for real geeks: I used to calculate every possibility explicitly, but I have replaced this with an excellent trick.

If the probability of winning Florida (27 electoral votes) is F and the probability of winning Ohio (20 electoral votes) is O, then the probability distribution for all the combinations (0, 20, 27 and 47 EV) can be expressed as the coefficients of the polynomial

[(1-F) + F * x^27] * [(1-O) + O * x^20], where x is just a placeholder variable.

You can see this by working out the example directly. As you can also see, when the number of possibilities is large (in the millions or more) this is far faster than going through each combination explicitly. In fact, it reduces the calculation from exponential time to polynomial time. (That last sentence was perhaps geekiest of all!)