That was me, I saw your math. I disagreed with it.

Here's the thread:
https://forums.civfanatics.com/threads/how-much-intelligent-life-is-there.302505/
Apparently awhile ago is almost 10 years. BTW, you also still have to reckon with that 10 year bet you lost in the cool unbuilt structures thread:
https://forums.civfanatics.com/threads/the-cool-unbuilt-structures-thread.196126/page-4
I agree that your new argument is better put, but I still think you're succombing to issues regarding selection bias. So let's reletigate for old times sake.
So let's say I'm God, and you getta be my archdemon (I'm an evil God

)
I'm sitting around on my throne and I get a bug up my butt and say. Gosh darn it, imma make me a big ol' batch of universes (let's say 10^75 of 'em).
So here's what I do, first I make a barren universe with 10^25 stars. Then for each star I roll a fair n-sided die, if it's 1 I put some life on it, if not I move on without the life.
But what dice should I use? I have in my dice drawer 10^50 fair dice a D1 a D2 a D3 all the way up to a D10^50. Well if I picked one at random that would almost certainly be greater than 10^48, thats' not very sporting. So instead here's what I do, I take a tape measure 50 units long and pick a random point on it x and measure it to divine precision. Then I take 10^x, round to the nearest integer and use that dice from the drawer. You'll note that this produces a distribution of n such that when written in scientific notation has a 2% shot of having any particular exponenent.
You'll note that that there are roughly 3 different types of batches I can produce:
If I'm using a D10^23 or less. Almost all the universes will have stars with life. Moreover almost all will have multiple stars with life. This is group has a 46% chance of occuring. (let's call this a "life-rich" batch)
If I'm using a D10^27 or more. Almost all the universes will be devoid of life. Moreover almost all with life will be the only ones in their universe with life. This is group has a 46% chance of occuring (let's call this a "life-poor" batch).
If I'm using a D10^23 to D10^27 things get complex. You'll see significant numbers of universes devoid of life, and with life. In this group a significant minority have only 1 star with life. This group has an 8% chance of ocurring. (let's call this a "intermediate" batch)
(I think this is a relatively fair distribution for the purposes of my argument though not a particularly realistic one. The salient properties of this distribution are that there is a significant possibility of the die being over and under the amount of stars in the universe with it being about the same a distinct but rare possibility. I believe that such a wide range of possibilities should be considered prior to attempting any statistical argument)
So let's say I whip up that batch of 10^75 universes and pick a universe at random. Then I ask you "what is the probability there is life in that universe?" Well if it's a life-rich batch you're almost certainly going to find life, if it's life poor it's almost certain you won't, in the intermediate batch there's a significant fraction of both. Your calculation is going to be about 50%. Now let's say I tell you, "actually in this universe there is life on at least one star" then ask "what do you think the probability of finding life on multiple stars is?". From this information you can pretty much rule out the possibility the batch is life-poor (because you probably wouldn't have found life on it), so you only only got life-rich and intermediate batches to worry about. Almost all the life rich ones have multiple stars with life. Most of the intermediate batch with life also have multiples, though a significant fraction don't. So overall you'd have to say the probability is something like 99% that there are multiple instances of life. I believe this is the sort of intuition you're using to come up with the above argument.
But what if instead of picking a universe at random after whipping up the batch I list all the stars with life and pick one at random, then ask you the same questions about the universe it's in? Well the first question "what is the probability there is life in that universe?" is pretty easy. It's 100% gaurenteed (ignoring the extraordinarily insignificant possibility that all universes are devoid of life) because if it didn't have life there'd be no stars on the list. Now if "what do you think the probability of finding life on multiple stars is?" the calculation is different. You don't get to rule out all those life-poor batches, because even though the chances in any particular universe is small, it's all but certain that that list is going to have at least some stars with life. So all options are back on the table again. So let's go through 'em. If it's a life-rich batch, you're probably going to find multiple instances. If it's a life poor batch, it's probably alone. And the intermediate batches have a mix of both. So you're going to come up with an answer of about 50% of multiple and 50% alone.
I argue that when assessing these probabilities we should be thinking about the latter not the former. The reason being that since we come from a star with life we will not find ourselves in some random universe from the batch, instead we will only find ourselves in universes that have at least one star with life. Without further information we should presume we're on a random star with life. If we are open to a significant possibility that dice has more sides than there are stars in the universe, then we must be open to the significant possibility that we are alone. I have thus far not seen any particularly solid argument that should preclude the probability of life being that low so accept that as a distinct possibility and with that accepting I accept that there's a significant possibility we're alone in the universe.