Beauty 's Cautionary Tale

The Problem Sleeping Beauty will participate in an experiment in which she will be put into a deep sleep on Sunday night. She will be awakened during the experiment either once or twice depending on the toss of a fair coin. In either case she will be awakened on Monday. She will be unaware upon awakening that it is Monday, but subsequently will be told that it is Monday and then given drugs to put her back to sleep. These drugs will erase her memory of having been awake. If the coin lands heads, Beauty will then sleep until the experiment ends on Wednesday. If it lands tails, she will be awakened briefly again on Tuesday, unaware of the Monday awakening. Beauty knows all of this, and will be able to distinguish waking up within the experiment from other awakenings. Before being put to sleep, it is reasonable for Beauty to assign a probability of 1/2 to the proposition that the coin lands heads. Now, let us assume that the experiment has begun and Beauty has just awakened from a deep sleep. What should her probability be now for heads? It seems that surely her probability should remain at 1/2. Beauty knew in advance that she would be awakened at least once, and so upon awakening receives no new information that could count as evidence for or against heads. If however the experiment were to be repeated, in the long run there be would twice as many tails awakenings as heads awakenings, and this suggests that the answer should be 1/3. Additional arguments have been proposed for both answers. Since the discussion has been largely a response to Elga (Elga 2000), who defends the answer of 1/3, I shall begin with his argument.