Is It Correct to Talk About the Probability of an Event that Has Already Occurred and Is Not Repeatable? I have a limited understanding of probability, PDFs, etc. When the Viking landers went to Mars, there was talk of the probability of life on Mars. Now we read of the probability of life beyond the solar system. It seems to me though, that there either is life beyond the solar system or there isn't. It's not like putting one's hand into a jar filled with black and white marbles in some proportion and taking one out. One might repeat that experiment many times and try to estimate the true proportion of black vs. white, or one might know the proportion beforehand and calculate the probability of pulling out 10 black and 5 white, or 15 all white, etc. I just have a hard time understanding what the "probability of life elsewhere in the universe" actually means.
 A: I think that there are objective and subjective probabilities. The objective probabilities are associated with physical phenomena such as radioactive decay, which is a stochastic process that has nothing to do with your presence at all. Whether you know about the decay or not, the decay process goes at its own pace, and has been going on before humans even existed. The rate of decay is related to the probabilities, objective ones.
The concept

probability of life elsewhere in the universe

could also be cast as the objective probability. For instance, we figured out how life originates, and we know enough about this Universe to figure out what is the probability that given this Universe life has originated in it.
However, this particular concept is not cast this way, because we don't know nearly enough about origination of life and this Universe. Hence, our statements regarding this  probability are based on our limited knowledge, and on what we know about the limits of our knowledge, thus we're talking about subjective probabilities. Subjectively, I may think that it's 50/50, and there is no way to prove or disprove me. 
The confusion is from that Bayes theorem. It works fine with objective probabilities, and it is the basis of the treatment of subjective probabilities too. That's why Bayesian people often don't understand the objective probabilities, they think all probabilities are subjective; they like to talk about "beliefs", for example.
UPDATE:
Monty Hall problem is an excellent example of subjective probabilities, and the power of Bayesian approach. You don't know where the money bag is, but it is certainly behind one of the three doors. So, when we attach a probability of 1/3 in the beginning, it is simply a statement about the state of our knowledge with regards to a money bag, and not about the actual location of the money bag. 
So, when you learn how Bayesian approach works on these problems, you tend to get carried away and start thinking that everything's relative to our knowledge. This is not the case with a real world. For instance, consider a wave function $\Psi(x,t)$ in quantum mechanics. The probability that a particle is in $x$ at time $t$ is $P=|\Psi(x,t)|^2$. This is not that we do not know where the particle is exactly, and that at some point we'll figure it out. No, this is an objective probability, that's how Nature works: it throws the dice.
A: There are some aspects that the otherwise excellent answer by @Aksakal do not touch.  Can we speak about probabilities after the event? In some cases, yes.  But do not expect much money if you run too the bookmaker's office and try to set money on the horse you just saw past the finishing line!  If you know what happened, then there are no more probabilities in play.  But you can throw a coin, catch it in the air and covering it with your other hand (without seeing). Then the result of the throw has been decided, but you do not know the result (nor do anybody else).  Then it gives just as much meaning to say "50% chance for tails" as it did before the throw.  What decides the probabilities (objective or subjective) is the state of knowledge, not the state of the world.
With philosophical jargon we could say that probability belongs to epistemology, not to ontology.  That answers your question about the "probability of life somewhere else in the universe".  Either it is or it is not, yes.  But we don't know! and that is what matters.  As for taking numerical estimates about that probability serious, that's another matter.
A: This is a Bayesian understanding of probability. For a Bayesian, probability reflects a degree of belief. When a Bayesian says, "There is a probability that..." they are saying, "I haven't excluded the possibility that..." They have no problem implicitly ascribing probability to events for which they've collected no data. To a Bayesian, excluding a possibility means that even seeing evidence to the contrary will not convince them otherwise. Practically it means there is no rationale for collecting data on the subject. So in an optimistic light, the claim as you mention is saying the opposite, "It is worth collecting data on whether or not intelligent life exists beyond our solar system."
