Nope, that’s not the model. Your initial expected population is 1111/4≈278. After the anthropic update, your probabilities of being in the boxes are 1/1111, 10/1111, 100/1111 and 1000/1111 (roughly 0.09%, 0.9%, 9% and 90%). The expected population, however is (11+102+1002+10002)/1111≈909. That’s an expected population update of 3.27 times.
Note that, in this instance, the expected population update and the probability update are roughly equivalent, but that need not be the case. Eg if your prior odds are 1:1:10−36 about the population being 1, 1000, or 1012, then the expected population is roughly 500.5, the anthropic-updated odds are 1:1000:10−24, and the updated expected population is roughly 1000. So the probability boost to the larger population is roughly (10−24/1000)/10−36=1021, but the boost to the expected population is roughly 2.
Nope, that’s not the model. Your initial expected population is 1111/4≈278. After the anthropic update, your probabilities of being in the boxes are 1/1111, 10/1111, 100/1111 and 1000/1111 (roughly 0.09%, 0.9%, 9% and 90%). The expected population, however is (11+102+1002+10002)/1111≈909. That’s an expected population update of 3.27 times.
Note that, in this instance, the expected population update and the probability update are roughly equivalent, but that need not be the case. Eg if your prior odds are 1:1:10−36 about the population being 1, 1000, or 1012, then the expected population is roughly 500.5, the anthropic-updated odds are 1:1000:10−24, and the updated expected population is roughly 1000. So the probability boost to the larger population is roughly (10−24/1000)/10−36=1021, but the boost to the expected population is roughly 2.