JeffJo comments on Probability is in the Mind - Less Wrong
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The random process a frequentist should repeat is flipping a random biased coin, and getting a random bias b and either heads or tails. You are assuming it is flipping the *same biased coin with fixed bias B, and getting heads or tails.
The probability a random biased coins lands heads is 1/2, from either point of view. And for nshepperd, the point is that a Frequentist doesn't need to know what the bias is. As long as we can't assume it is different for b1 and 1-b1, when you integrate over the unknown distribution (yes, you can do that in this case) the answer is 1/2.