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JGWeissman comments on Outlawing Anthropics: An Updateless Dilemma - Less Wrong
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What if, instead of requiring agreement of all copies in a green room, one copy in a green room was chosen at random to make the choice?
In this case the chosen copy in the green room should update on the anthropic evidence of being chosen to make the choice. That copy had a 1/18 probability of being chosen if the coin flip came up heads, and a 1/2 probability of being chosen if the coin flip came up tails, so the odds of heads:tails should be updated from 9:1 to 1:1. This exactly canceled the anthropic evidence of being in a green room.