  The lack of evidence for the existence of extraterrestrial life, even the
simplest forms of animal life, makes it is difficult to decide whether the
search for extraterrestrial intelligence (SETI) is more a high-risk,
high-payoff endeavor than a futile attempt. Here we insist that even if
extraterrestrial civilizations do exist and communicate, the likelihood of
detecting their signals crucially depends on whether the Earth lies within a
region of the galaxy covered by such signals. By considering possible
populations of independent emitters in the galaxy, we build a statistical model
of the domain covered by hypothetical extraterrestrial signals to derive the
detection probability that the Earth is within such a domain. We show that for
general distributions of the signal longevity and directionality, the mean
number of detectable emitters is less than one even for detection probabilities
as large as 50\%, regardless of the number of emitters in the galaxy.
