[jonsen]

... because each coin is equally as likely to be selected from the jar

That's not the reason. It's because the fake coin also has two sides and therefore also 2^11 different outcomes. It just happens they all look the same.

Otherwise I agree with your argument.

[baddox]

True, I could have still used zeros and ones to enumerate all outcomes of the biased coin, but said that both digits represent heads.