That said the subject we're discussing isn't a simple game of cards, dice, or coins — Agent Smith
True. If you want to quantify probability then you need to compare one thing with another. One way to quantify a comparison is to express it as a proportion. So you can quantify the probability that we'll see aliens in the next ten years just as you can quantify the probability that you'll catch a cold in the next two years. The difference is that in one case you have some data to start with. In the other you have just speculation. The answer you get out will depend entirely on the assumptions you put in.
Let's say that there's a 50% chance that aliens will be observed on some day in the next ten years. You can work backwards from the calculation I gave above and work out the probability that, on any given day, aliens will be observed. That calculation is fine. But you will have made the whole thing up. The 50%, the ten years, and therefore the daily probability will be entirely speculative. It might be 50% in ten years or in fifty or a hundred.
It's like this with the aliens:
The probability that on a given day we'll see aliens is p. 0 < p < q.
The entirely made-up probability that we'll see them in the next ten years is 50%. It could be 30% in fifty years or 90% in five hundred. We are just guessing. But let's stick with 50% in ten years. It's nothing to do with equilibrium and agnosticism, because the choice of the number of years, ten, twenty or a hundred, will affect the calculation and we can be as agnostic about five hundred years as about five years.
There are roughly 3,650 days in ten years.
Now, the probability that we won't see aliens is 1 - p.
So the probability that we won't see aliens on any day in ten years is (1 - p) ^ 3,650.
Now, we've decided arbitrarily that (1 - p) ^ 3,650 is 50%, that is, it's as likely that we'll see aliens as not in ten years.
What is the probability that we'll see aliens on at least one day? Answer: about 0.00019.
Note: that figure, 0.00019, is entirely the result of an arbitrary choice about how long we must wait for us to have even chances of seeing or not seeing aliens. It has nothing to do with the likelihood of seeing aliens. It has proceeded entirely from our speculation.
We have no data. And we can make up data. But we cannot then say anything about the probability of any event.
So, yes, mathematical probability can be assigned to any supposed event. But in the absence of data, we are just assigning whatever value we want. It's fantasy and fiction and not to be confused with truth in any way whatsoever.