Does the drake equation include earth?

It means 2.31 total in the galaxy. So we're one of them and the probability (according to this estimate) is that there are "1.31" others.

But don't read too much into this number. There are still a lot of unknown variables in the Drake equation.

The thing to keep in mind about the Drake equation is that it's a VERY rough estimate - it relies on variables that we don't know the values of very well, and because it's dealing with probabilities, it's really better to express the result as a range - something like, "there's a 95% chance that the number of civilizations in the galaxy is between 1 and 4". (Note, I don't know that's the case, I'm just making up numbers!)

Given the somewhat murky nature of the Drake equation, I'd say that it's fair to interpret it either way. Say there are 300 billion stars in the Milky Way. If you're an observer from some other galaxy, and you don't know Earth exists, and you calculate the number of civilizations there are likely to be in the Milky Way, you'd say, "I expect to find 2.31 civilizations among these 300 billion stars." (Ignoring for the moment that you can't have 0.31 of a civilization!)

Observing from Earth, you say, "There are 299,999,999,999 other stars in the Milky Way. My presence here on this one star doesn't affect the odds of life elsewhere in the galaxy. So I expect to find 2.31 * 299,999,999,999 / 300,000,000,000 ~= 2.31 civilizations among all those other stars."

So, since we know ahead of time that there's intelligent life here on Earth, I'd say we're in the 2nd position, and that we expect to find 2.31 other civilizations in the Milky Way.

I hope that helps!

The best way to understand the results of the Drake equation (whether you accept them or not) is to treat them as an "average number per galaxy". So when you do apply this number to our own Galaxy, then it includes us.

The results of the Drake equation still have a very wide variance. Originally, results in the hundreds of thousands were not rare. Then, as scientists began to understand ALL the conditions that are necessary for life to exist AND to evolve to an intelligent stage, the numbers kept going down, reaching a low in the 1970s (I remember seeing a result of 0.25 per galaxy).

In this particular case, if you apply this number to our Galaxy, you would interpret this result as saying: we are lucky to be here. Mathematically, you could interpret it as meaning: You should expect to find one civilization for every 4 galaxies that you survey.

SInce our Local Group has three spiral galaxies (ours is one), plus a bunch of dwarf galaxies, the result could be interpreted as meaning: "there is a possibility that we are the only civilization in the Local Group".

One problem with statistical results is that they are difficult to apply to a single object. If the result is 2.31, this means that if you survey more and more galaxies (hundreds) you should find that the number of civilizations per galaxy should tend towards 2.31

That is all it really means.

And, in this calculation, ours would definitely count as 1.

The next important number, in a statistical analysis, is the variance. Even if the average were to be 2.31, does that means that we should ALWAYS expect to find at least one (or even two) per galaxy? The Drake equation does not really help us there.

I suspect that if we did survey a galactic super-cluster, the distribution of results (the actual number of civilizations per galaxy) would appear to be a "Poisson distribution", with some galaxies having 0, the peak number of galaxies with civilization would have 2, with the next highest frequency of results being 3.

If the supercluster were large enough, the probability of finding a galaxy with a much higher number of civilization would get smaller and smaller, but the probability never goes down to zero.

You could, for example, find one galaxy with 10 civilizations, and that would still fit with the average being around 2.31

Of course, we have not conducted this survey yet. And many of the numbers used as factors in the Drake equation are still guesses. They may be better guesses than they were forty years ago, but they are still guesses (as we have not yet been able to go out and check our results).

Your result is not correct. There are unknown variables in the Drake equation which give a broad range of answers, depending on what value you assign to them. For example, the average number of planets in each solar system capable of supporting life is completely unknown. Most of the variables are unknown, and have to be merely guessed at.