The universe is an awfully big place. Granted, most of it is empty space. But within that empty space, there are trillions of stars. Maybe more. At least some of those stars have planets around them, and some of those planets are the kind that could give rise to life. That's still hundreds of billions (at least) of planets that can support life out there. And so, the popular argument goes, even if the odds of life arising on another planet are very small, there are so many planets that it is bound to happen. Thus, there almost certainly exist extraterrestrial life forms. It isn't a matter of if, but of when we find them.
Here's a video of Dr. Carl Sagan presenting a more sophisticated version of this (with actual numbers) to estimate the number of inhabited planets in our galaxy. Or try this worksheet on the Drake Equation on the BBC website.
It's a common argument. And it sounds pretty convincing. If you keep trying over and over, even though something is unlikely, eventually you will succeed.
It's common and convincing, but it's also fallacious. Here's the problem: How many times do we have to try before we're guaranteed to succeed?
The mathematical answer is infinitely many times.
