By Nick Bostrom
Department of Philosophy, Oxford University
IV. THE CORE OF THE SIMULATION ARGUMENT
The basic idea of this paper can be expressed roughly as follows: If there were a substantial chance that our civilization will ever get to the posthuman stage and run many ancestor-simulations, then how come you are not living in such a simulation?
We shall develop this idea into a rigorous argument. Let us introduce the following notation:
: Fraction of all human-level technological civilizations that survive to reach a posthuman stage
: Average number of ancestor-simulations run by a posthuman civilization
: Average number of individuals that have lived in a civilization before it reaches a posthuman stage
The actual fraction of all observers with human-type experiences that live in simulations is then
Writing for the fraction of posthuman civilizations that are interested in running ancestor-simulations (or that contain at least some individuals who are interested in that and have sufficient resources to run a significant number of such simulations), and 
for the average number of ancestor-simulations run by such interested civilizations, we have
and thus:
(*)
Because of the immense computing power of posthuman civilizations, is extremely large, as we saw in the previous section. By inspecting (*) we can then see that at least one of the following three propositions must be true:
(1) 
(2) 
(3) 
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