It is impossible to actually calculate the probability of detecting extrasolar civilizations, simply because too many of the variables involved are unknown. However, there are some general guidelines which can help us explain why, at the very least, detecting extrasolar civilizations has so far proved so difficult (i.e. to the best of public knowledge, it hasn't happened yet). These involve figuring out how likely it is that extrasolar civilizations actually exist, and then figuring out what the best way of finding them might be. Overall, it is still only uncertain whether extrasolar civilizations exist (though they probably do), and even less clear whether our current technology actually gives us a reasonable means of detecting them.
- Are They Out There? -
Two speculative "principles" guide us in understanding the probability that there are extrasolar civilizations which we could detect. The first, the Drake Equation, is an attempt to estimate how many civilizations should exist in the galaxy, based upon factors we think are essential for a spacefaring civilization to develop: a planet roughly like ours, orbiting a star roughly like our Sun, on which primitive life first forms, and then gradually evolves increasing complexity until an intelligent civilization emerges - all without the process being interrupted by too large an asteroid impact, or a gamma ray burst from a nearby star baking everything on the planet in question.
The odds are astronomical - which is okay, since we're dealing with astronomical numbers here. The Drake equation shows that, given all but the most pessimistic assumptions about how life evolved on Earth, there should be at least several intelligent civilizations active in the Milky Way Galaxy at any one time. An optimistic but not foolishly idealistic input to the Drake equation yields a prediction of about five to twenty intelligent civilizations in our galaxy at any one time, assuming that there is a continuous process of civilizations arising, growing, and then declining and going extinct.
The second principle, however, throws some cold water on this optimism. If the odds are that many spacefaring civilizations exist, says the Fermi Paradox, why haven't we seen them yet? In a little over a century, humanity has progressed from hot-air balloons to the Space Shuttle; we have gone from primitive submarines to space probes that are even now at the edges of our solar system. Assuming this is an average rate of progress once a civilization reaches a certain level, and assuming that it continues more or less indefinitely, then an interstellar civilization just a few million years older than us should be visible everywhere in our galaxy. Unless they have gone extinct already - or unless, of course, they were never there to begin with. A variety of speculative solutions have been proposed for the Fermi Paradox, explaining that perhaps advanced alien cultures are uninterested in exploration of the physical universe or have deliberately chosen to conceal themselves, but, given the evidence, the simplest explanation is that they don't currently exist.
That something is holding back life from becoming the massive interstellar civilizations seen in the more optimistic science fiction, like Star Trek, is a disturbing thought. However, it is always important to remember that the search for extrasolar civilizations might always turn up empty. We could be the only intelligent civilization close enough to ourselves that we will ever know for certain actually exists.
- Can We Detect Them? -
Assuming for the moment that extrasolar civilizations exist out there at the present time, the next question is how to go about detecting them. The current standard model for the Search for Extraterrestrial Intelligence (SETI) holds that we can find alien civilizations by listening for their radio transmissions, with the assumption that they have followed a path of technological development similar to ours and are therefore emitting a large number of essentially uncontrolled transmissions from their equivalent of television, radio, and other telecommunications devices. So far, various SETI projects around the world have failed to detect the telltale patterns of artificial broadcasting by listening to samples from radio telescopes like the Arecibo Observatory in Puerto Rico. However, because of the expensive telescopes and massive computing resources required, so far all of our SETI efforts amount to a brief snapshot of a fraction of the night sky. As SETI resources increase, the likelihood that we will be listening at the right angle, at the right time, does increase.
Again, however, if long-lived civilizations existed, they should be producing many such transmissions from many different planetary sources - unless, of course, interstellar travel is as impossible as contemporary science states, rather than as relatively simple as much popular science fiction assumes. The fact that we are having difficulty finding these transmissions indicates a good chance that extrasolar civilizations either do not exist, or are not communicating the way we expect them to.
If it's the second choice, then the failure is one of imagination and technology on our part. A variety of speculative explanations have been put forward on this count. For example, it is possible that extrasolar civilizations choose to be silent, in fear of attracting hostile attention (which is famed physicist Stephen Hawking's recommendation for Earth, as well). It is even conceivable, if barely, that we have been identified as a primitive civilization and somehow separated from the main current of interstellar activity. It is more credible, if more difficult to imagine, that highly advanced extrasolar civilizations have simply invented some form of sophisticated communications technology far beyond mere radio waves. In that case, since we're effectively listening on the wrong channels, we will never hear anything.
In short, it is impossible to calculate the probability of detecting extrasolar civilizations, but given our failure thus far as well as some of the conceivable problems in doing so, it is reasonable to assume that our likelihood of doing so is not very high.