I asked my caveman friend to translate. He's a paleoanthropics expert.
Big chunk of space! It has three parts. Is there some guy in each part? Let's say no. Not unless the part is very lucky!
Now think of many such chunks of space that could have been! Whoa! Sense of wonder! Let's pick some guy in some chunk. That'll be us!
First let's pick some random chunk. Self-Sampling Assumption says we're a random guy in the chunk! (What if there is no guy in the chunk? Don't think about it!) Are we alone? Probably yes! Most chunks with a guy don't have a second guy. Because we said guys are rare! Math!
But now let's not pick a random chunk. Let's pick a random guy, in any chunk. Say there's two guys in a chunk. Then we'll pick a guy in the chunk twice as often! Self-Indication Assumption! (Maybe they meet and live happily ever after. Just because I'm caveman doesn't mean I heteronormatize!) Now are we alone? Still probably yes! Most guys are in their own chunk. Yes, if there's two guys in a chunk it has two chances to be picked. But there's just so few chunks with two guys. Because we said guys are rare! So this Self-Indication business hardly matters at all! Math!
That could have saved me so much time. Thankyou!
Consider a scenario in which there are three rooms. In each room there is an independent 1/1000 chance of an agent being created. There is thus a 1/109 probability of there being an agent in every room, a (3*999)/109 probability of there being two agents, and a (3*9992)/109 probability of there being one.
Given that you are one of these agents, the SIA and SSA probabilities of there being n agents are:
The expected numbers of agents is (1(3*9992) + 2(2*3*999) + 3(3*1))/(3*1+2*3*999+1*3*9992) = 1.002 for SIA, and (1(3*9992) + 2(3*999) + 3(1))/(1+3*999+3*9992) ≈ 1.001 for SSA. The high unlikelihood of life means that, given that we are alive, both SIA and SSA probabilities get dominated by worlds with very few agents.
This of course only applies to agents who existence is independent (for instance, separate galactic civilizations). If you're alive, chance are that your parents were also alive at some point too.