The Doomsday Argument
This puzzle, or seeming paradox, has no simple solution.
The underlying problem relies on the assumption that we, as observers, should not assume that there is anything unique about our position unless there is evidence to do so. This is known as anthropic reasoning. (Based on the commonly used term "The Anthropic Principle")
For example, consider the fact that the night sky seems more or less uniformly filled with stars in all directions. This would be consistent with the Earth being at the center of the universe. But seeing that any star might form planets, and life could have (and in fact may have) evolved on any planet, we should likely conclude that we might have evolved on any planet in the universe. Therefore, it is statistically extremely unlikely that we would find ourselves in the dead center of the universe.
And so scientists have long since abandoned the notion that we are in the center of the universe (in fact, modern cosmology holds that the universe doesn't even have a center). Note that anthropic reasoning could only be applied once we knew that other star systems existed.
Now back to the problem. If this logic holds true for all observers, than everyone should say, "There's nothing special about my position, I could be any observer."
But what does "any observer" mean? Obviously in the case of this problem "I" could not have been an E Coli bacterium. A bacterium lacks consciousness, and it certainly lacks the mental faculties to understand this problem.
So we're forced to clarify our principle with something called the "reference class": when we say we could have been any observer, we must qualify that "any observer" must have similar mental states, similar enough that s/he would be able to conceptualize the same reasoning. After all, 99.99999999% of the life forms on Earth are single-celled, but no one is staggered to learn that they are multi-celled, despite the odds against any random life-form being multi-celled.
So we define our "reference class" as the group of observers who are similar enough to us to have the same perspectives. Unfortunately, deciding how narrowly to define our reference class is rather tricky. Is it all intelligent life in the Universe that will ever live? Is it every human? Every human since language evolved? Every human since the Doomsday Argument was formulated? Just me?
Rather than go through all the various comments that have been made on this problem, I'll give a link to an excellent overview of the Doomsday Argument and the Anthropic Principle..