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I am reading Bostrom's famous paper "Are you living in a computer simulation?", but I can't understand his probabilistic reasoning.

Why does he define the fraction of observers that live in a simulation in the following way?:

fsim = fp * N * H / (fp NH) + H?

And why does it then consider just fi as the numer of posthuman civilizations interested in the simulations?

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Bostrom's idea here is based on several assumptions, for example:

  1. there were a substantial chance that our civilization will ever get to the posthuman stage and run many ancestor‐simulations with enormous amounts of computing power available by then.

  2. you have no information that bears on the question of which of the various minds are simulated and which are implemented biologically.

So following his paper referenced here:

fp : Fraction of all human‐level technological civilizations that survive to reach a posthuman stage

N : Average number of ancestor‐simulations run by a posthuman civilization

H : Average number of individuals that have lived in a civilization before it reaches a posthuman stage

H is the average number of biological individuals that have lived in a civilization before it reaches a posthuman stage including those civilizations which haven't been able to reach posthuman stage, thus your denominator (fp NH) + H is the number of total individuals biological and simulated combined per civilization on average. And your numerator (fp NH) is just the average number of simulated individuals per civilization on average. So naturally your whole formula describes the actual fraction of all observers with human‐type experiences that live in simulations. After easily eliminating H for both numerator and denominator and based on assumption 1) above, it's easy to see fsim approaches to 1 if fp is not approaching 0.

Then he substitutes N=fi*Ni to further refine his conclusion to 3 possible propositions, otherwise he can only arrive at 2 possible propositions and without (2).

fi: the fraction of posthuman civilizations that are interested in running ancestor‐simulations

Ni: the average number of ancestor‐simulations run by such interested civilizations

... we can then see that at least one of the following three propositions must be true:

(1) fp ~= 0

(2) fi~= 0

(3) fsim ~= 1

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