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AlephNeil comments on Conditioning on Observers - Less Wrong
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I do like this intuition pump - it gets across how weird the situation is - but here's one thing you may not have realised:
Once the experiment is over, then even based on thirder logic, people ought to cease saying "I'm 90% sure..." and start saying "I'm 50% sure..."
(Because they have forgotten the 'extra information' that led them to give 90% rather than 50% as their probability of cancer.)