The Sleeping Beauty Problem

Sleeping Beauty participates in the following experiment. On Sunday she is put to sleep. A fair coin is flipped. If it lands Heads, she is woken on Monday only. If it lands Tails, she is woken on Monday and again on Tuesday (with her memory of the Monday awakening erased before Tuesday). Each time she wakes, she is asked: 'What is your credence that the coin landed Heads?' She knows the full protocol. (a) Present the argument that her answer should be $1/3$ (the 'thirder' position). (b) Present the argument that her answer should be $1/2$ (the 'halfer' position). (c) Suppose the experiment is repeated 1000 times independently. If Beauty bets $1 at even odds on Heads each time she is woken, what is her expected net gain or loss over all awakenings? What does this imply about the two positions?