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DanielLC comments on Jokes Thread - Less Wrong Discussion
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Not so fast.
Not quite. In your example 0.2 of coins are not biased, they are predetermined in that they always provide the same outcome no matter what.
Let's try a bit different example: the prior is that 10% of coins are biased towards heads (their probabilities are 60% heads, 40% tails), 10% are biased towards tails (60% tails, 40% heads), and 80% are fair.
After one throw (let's say it turned out to be heads) your posterior for the fair coin did not change, but your posterior for the heads-biased coin went up and for the tails-biased coin went down. Your expectation for the next throw is now skewed towards heads.
I didn't realize you were serious, given that this is a joke thread.
Here's the easy way to solve this:
By conservation of expected evidence, if one outcome is evidence for the coin being biased, then the other outcome is evidence against it.
They might believe that it's biased either way if they have a low prior probability of the coin being fair. For example, if they use a beta distribution for the prior, they only assign an infinitesimal probability to a fair coin. But since they're not finding evidence that it's biased, you can't say the belief is based on the outcome of the toss.
I suppose there is a sense it which your statement is true. If I'm given a coin which is badly made, but in a way that I don't understand, then the first toss is fair. I have no idea if it will land on heads or tails. Once I toss it, I have some idea of in which way it's unfair, so the next toss is not fair.
That's not usually what people mean when they talk about a fair coin, though.