I've had some fun thinking about these (and related) problems, but things get very complicated very quickly, and I constantly doubt my own work! I would like to see what others do with them. Thanks!
You have two identical-looking coins, but one is fair, and the other is unfair, coming up Heads 2/3 of the time. You flipped coin A once, and it came up Heads. You flipped coin B three times, and got two Heads and a Tail.
a) What's the probability that coin A is the unfair one?
b) Suppose you were allowed to perform one more flip, then declare which coin you believed to be unfair. Which coin should you flip? After performing this flip, which coin will you say is the unfair one (based on the result of that flip), and how likely are you to be correct?