The first possibility, that we just happened to luck out (being around for the first and only alien encounter), is less probable than that you not someone, but you will win next months lottery jackpot. It strains credulity, to use polite vernacular.
The second possibility, that Earth hosts extraterrestrials on a routine basis, and therefore a visit during your lifetime is not particularly improbable, deserves a bit more scrutiny. The question is, how often do they visit? If its only once in a few tens of millions of years, were back to the first possibility, and the odds are highly stacked against you being one of the lucky visitees. But some folk claim that aliens have glissaded to Earth in historical times (five millennia ago, when the pyramids were built, or one millennium ago, when the Nazca Indians elected to decorate the Peruvian desert floor with glyphs of turkeys and other of their favorite fauna). If any of this is true, it argues for visits at least once every 1,000 years or so. The problem with this is that barring some reason for them to visit humans in particular (a possibility we consider below) it implies that there have been millions of expeditions to Earth! We may send the occasional anthropological research team to Borneo, but we dont send millions. And its a lot easier to get to Borneo than to traverse hundreds or thousands of light-years. This, too, seems to be an unlikely explanation for visitors now.
Finally, we consider door number three were we have enticed the aliens with human activity. Lets set aside the question of whether advanced galactic societies would have the slightest interest in our wars, our pollution problems, or our reproductive systems. The real question is, how would they know about us at all?
In fact, theres only one clear and persistent "signal" that Homo sapiens has ever sent to the stars: our high-frequency radio transmissions, including television and radar. The Victorians (let alone the Egyptians or the Nazca Indians), despite all their technical sophistication, could never have been spotted from light-years away. Humans have been making their presence known to the universe only for the last 70 years or so.
And thats a problem. It means that even if, after receiving an earthly transmission, the aliens can immediately scramble their spacecraft and fly to Earth at the speed of light, they cant be farther than 8 light-years away to have arrived by 1947. There are four star systems within this distance. Count em, four. Were back to winning the lottery.
Dont get me wrong.I understand that some people love the idea of Earth or its human inhabitants have not only attracted the attention of galactic neighbors, but encouraged them to visit. But frankly, the numbers dont give much support to this somewhat self-indulgent idea.