You're both welcome. A couple of things more.
First, I just wanted to add that Fermi was no idiot, of course. But the assumptions he made date back to 1950. We know much more about planets now. Something we know now, for example, is that the Earth-Moon system is far from typical. The Moon is an unusually large satellite and has a rather bigger than normal stabilizing effect on the Earth's tilt. To the point that astronomers are starting to look upon the Earth-Moon system as a binary planetary system, rather than a standard planet and its small satellite or group of satellites. The huge tidal effect that the Moon has on the Earth is believed to have played a major part in the origin of life at least during the first billion years, stirring the chemicals dissolved in water and thus triggering volume reactions (much quicker and efficient) rather than surface effects.
Another factor is the presence of outer giants like Jupiter and Saturn, for billions of years playing the role of shuttles for asteroids from the Kuiper belt, etc. Water and amino acids in the asteroids are also thought to have been very important. In case any of these factors were found to be essential to the appearance of life, it could be a basis to estimate the number of solar systems in the Milky Way that satisfy similar conditions. Would other different sets of conditions be just as good, or maybe even better? I don't know. I don't know if anybody knows.
Drake's equation came later than Fermi's argument (in 1961). Actually, I think Drake's equation is a more promising ground for estimating the chance of there being intelligent life forms, among other things, precisely because, although ambitious, it's a much less assuming parametrization of the probability, rather than an equation or a "closed" calculation. There is room for re-estimating the factors as we learn more about the phenomenology of galactic (or extra-galactic) solar systems. Plus the last factor is, if I'm not mistaken, the probability that a civilization will be able to send signals, rather than travel to Earth, which significantly increases the odds. Fermi was concerned with interstellar travel, AFAIK.
The detection of signals with a message in them will probably be the first evidence, if there ever is one, of some form of intelligent life besides us in the universe, rather than the flight of UFOs.
But here's the bad news, IMO:
Take a look at this table with time gaps separating the appearance of new levels of organization:
First prokaryotes (from Earth's formation): 1 billion years
First eukaryotes (from prokaryotes): 800 million years
First multicellular eukaryotic organisms (from single-celled eukaryotes): 2 billion years
First intelligent life (from multicellular eukaryotes): 700 million years
Average for the appearance of a new level of organization: 1.125 billion years
Now suppose there's a planet out there with something like eukaryotes (cells with a nucleus). You're going to have to sit there waiting for 1.1 billion years for you to see anything interesting to happen if the above table is anything to go by. That's the problem.