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NancyLebovitz comments on Open Thread, November 8 - 14, 2013 - Less Wrong Discussion
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I think you're assuming that Alice has to pick H or T randomly and then ask the third party if it's correct. But she doesn't have to do that. She can just ask the third party whether it's H, each time. Over time it will be confirmed that the coin is fair.
Yes, but my point was that her knowledge that the coin is fair doesn't help her improve her guesswork on the text toss over Bob, and someone judging her on the basis of her toss-by-toss successes wouldn't be able to ascertain that she has more accurate beliefs than Bob...