A coin flip she can't remember produces two competing 'correct' probabilities

Sleeping Beauty is put to sleep; a coin flip decides whether she's woken once, or woken twice with her memory erased between wakings. Each time she wakes, what should she believe the chance of heads was? Philosopher Adam Elga argued for 1/3, since two of the three possible wakings follow tails; others insist a fair coin's odds must stay at 1/2. Both camps have serious, unresolved arguments.

— Adam Elga, Self-locating belief and the Sleeping Beauty problem — Analysis, 2000