This is a bit of a preview of a paper my friend Russ Zaretzki are working on.
Take a gander at his pic:
This is the probability that John McCain wins the election given only the historical evidence of Republican/Democrat elections, and the fact that there will be just 1, 2, …, up to 38 more Republican/Democrat elections. Let me explain.
Since Democrat James Buchanan ran against Republican John C. Fremont in 1857, United States presidential elections have been dominated by these two parties. From that first contest, Democrats have won 16 elections and Republicans 22. This year we have another election in which the two parties are again featured. Now, this means that the number of elections of this type has so far been finite, and history strongly suggests that this series of elections itself will be finite; that is, some day it will not be Democrats versus Republicans, or even might even be that there will be no elections1.
How many more elections there will be is, of course, an open question. But let us suppose that the one before us is the last election between the two parties. Then, conditional only on the past elections, the probability that the victor will be a Republican is 0.577. The standard Bayesian (continuous-value approximation) estimate gives 0.575. The classical guess is 0.579.
Our new method of guessing is based on knowing that the number of elections has been and will continue to be finite, that is, that it will not be without number, going on forever. It is important to recognize that traditional methods make this assumption. That is, that the number of “trials” (elections) will be infinite.
Ok, ok. These don’t seem like very big differences—and for this problem, they are not. But let’s suppose that instead of this being the final election, we’ll have two more. Then the probability McCain wins is just over 0.575. If we think there will be 9 more elections, then the probability McCain wins this one is only 0.570. Once the number of future elections becomes “large”, our guess matches the standard Bayesian one. That’s what the dashed, black horizontal line is. The red dot-dashed line is the classical estimate.
Eh, not a very big difference either, but it could be enough of one if you were, say, making a bet. And in some other problems, the differences are enormous; but this problem is a lot more fun.
The probability is over 50%. It obviously does not account for anything except previous elections. But it’s enough to raise a smile.
Incidentally, the math for all this is very heavily related to Laplace’s probability of succession. Google that. We introduce a twist that makes solving it sensible for certain problems. The surprise is that the probability depends on knowing the future number of trials (that’s the big difference).
1 Ever notice that at the Democrat rallies you hear “Obama! Obama! Obama!”, while at the Republican ones you hear “USA! USA! USA!”?