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In discussions of the range of SETI, I'll often see it spoken of in terms of how far SETI could detect a signal like a given type, emitted by our civilization (e.g. Arecibo could detect a UHF TV carrier at .3 light years, the SVTC virtual telescope could detect an airport radar at 8 parsecs and BMEWS radar at 1000 parsecs, etc.) I'm wondering, though, how easy or difficult it would be to detect a civilization like ours in practical terms. Say, for instance, there was a civilization roughly like ours on a planet around Alpha Centauri. Based on what I've read, radio and television signals from such a planet would not be detected, but radars could. But how likely would it be that a detectable "leakage" signal like that was actually intercepted by SETI? It is my understanding that individual stars are only examined for short periods of time, and that powerful commercial signals (e.g. radars) are usually at least somewhat directional. So, we would have to be somewhat lucky to be listening to a given star at the right time, when the solar system is in the cone of a detectable signal, wouldn't we? How lucky would we have to be, in practical terms, for an unintentional leakage signal like that from a nearby star to be detected? If there were a civilization something like our own (maybe allowing for practices that make them somewhat less detectable, e.g. no large military radars and less reliance on air travel, but within what would be "reasonable" for an advanced civilization not deliberately trying to hide itself), around a nearby star (say, Alpha Centauri or Epsilon Eridani), would it almost certainly have been detected by now, or would there be a significant chance would would not have been lucky enough to have a radio telescope listening to that star at the right time? Deep Thought
The Doctor Responds: Let's assume that there are, say, 10,000 civilizations existing in the Milky Way galaxy right now, that are at the right level of technology for their electromagnetic emissions to be detectable by us. (That figure is, in fact, Frank Drake's own estimated "solution" to his famous Equation.) Let's further (and arbitrarily) assume that there's a oneinathousand chance that we actually will detect such a signal, given a reasonable period of SETI observations. (I'm not going to address just what a "reasonable period" might be  that's a subject for a separate analysis.) Now, what are the true odds of SETI success, given those figures? One (simplistic) school of thought holds that our odds are simply one in a thousand, or 0.1%. However, that is the probability of detecting any one of those 10,000 signals, and, even if we fail to detect that one, there are still 9,999 more to go So, this is obviously not the correct answer. Another (equally simplistic) solution would be to say "well, there are 10,000 of them, and the odds of detecting any one are 1,000 to one, so that means we'll ultimately detect ten civilizations." This solution is also incorrect, because it overlooks a fundamental mathematical principle: probabilities have no memory. Well then, just how should we approach this problem? We could turn the question around, and ask, "what is the probability that we won't detect any of those 10,000 signals?" This is a question we can in fact answer, and if we know the probability of failure (let's call it Q), we can then get the probability of success (I'll call this P). This is true because, for any experiment, there are only two possible outcomes: success or failure, P or Q. So, the probability of one or the other (either success or failure, P or Q) has to be 100%, or a ratio of 1. In other words, P + Q = 1. Always. So, all we have to do is calculate Q, and then subtract it from 1, to get P. How would we go about that? Here's one way: The odds of detecting any one signal, according to how we structured the problem, are 1 in 1000, for a probability of 0.001. So the probability of not detecting it would have to be 0.999. If there were two (completely independent) signals out there, the probability of not detecting either of them would be (0.999) squared, or 0.99801. If there were three signals, the probability of not detecting any of them becomes (0.999) cubed, or 0.997, and so forth. You can see that, the more signals there happen to be out there, the lower becomes the probability of complete failure. After all, to "succeed", SETI needs to detect only one civilization, not all of them. The general solution for outright failure, given Q (the probability of failure for one signal) and N signals, is Q^N. So, the probability of SETI success, Pn, is found from (1Q), and becomes:
Plugging in 0.999 for Q, and 10,000 for N, we get
= 0.999955 This means, given our assumptions, the chance of detecting any one of those assumed 10,000 signals, even if the odds of detection for each one equals 0.1%, would be 99.9955%! So, if we listen long enough, we are almost guaranteed SETI success. So, now, the only remaining question is, "how long is long enough?" I can't say, but obviously we aren't there yet!

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