“What is your credence now for the proposition that the coin landed heads?”
There are three doors. Two are labeled Monday, and one is labeled Tuesday. Behind each door is a Sleeping Beauty. In a waiting room, many (finite) more Beauties are waiting; every time a Beauty is anesthetized, a coin is flipped and taped to their forehead with clear tape. You open all three doors, the Beauties wake up, and you ask the three Beauties The Question. Then they are anesthetized, the doors are shut, and any Beauties with a Heads showing on their foreheads or behind a Tuesday door are wheeled away after the coin is removed from their forehead. The Beauty with a Tails on their forehead behind the Monday door is wheeled behind the Tuesday door. Two new Beauties are wheeled behind the two Monday doors, one with Heads and one with Tails. The experiment repeats.
You observe that Tuesday Beauties always have a Tails taped to their forehead. You always observe that one Monday Beauty has a Tails showing, and one has a Heads showing. You also observe that every Beauty says 1⁄3, matching the ratio of Heads to Tails showing, and it is apparent that they can’t see the coins taped to their own or each other’s foreheads or the door they are behind. Every Tails Beauty is questioned twice.
Every Heads Beauty is questioned once. You can see all the steps as they happen, there is no trick, every coin flip has 1⁄2 probability for Heads.
There is eventually a queue of Waiting Sleeping Beauties with all-Heads or all-Tails showing and a new Beauty must be anesthetized with a new coin; the queue length changes over time and sometimes switches face. You can stop the experiment when the queue is empty, as a random walk guarantees to happen eventually, if you like tying up loose ends.
“What is your credence now for the proposition that the coin landed heads?”
There are three doors. Two are labeled Monday, and one is labeled Tuesday. Behind each door is a Sleeping Beauty. In a waiting room, many (finite) more Beauties are waiting; every time a Beauty is anesthetized, a coin is flipped and taped to their forehead with clear tape. You open all three doors, the Beauties wake up, and you ask the three Beauties The Question. Then they are anesthetized, the doors are shut, and any Beauties with a Heads showing on their foreheads or behind a Tuesday door are wheeled away after the coin is removed from their forehead. The Beauty with a Tails on their forehead behind the Monday door is wheeled behind the Tuesday door. Two new Beauties are wheeled behind the two Monday doors, one with Heads and one with Tails. The experiment repeats.
You observe that Tuesday Beauties always have a Tails taped to their forehead. You always observe that one Monday Beauty has a Tails showing, and one has a Heads showing. You also observe that every Beauty says 1⁄3, matching the ratio of Heads to Tails showing, and it is apparent that they can’t see the coins taped to their own or each other’s foreheads or the door they are behind. Every Tails Beauty is questioned twice. Every Heads Beauty is questioned once. You can see all the steps as they happen, there is no trick, every coin flip has 1⁄2 probability for Heads.
There is eventually a queue of Waiting Sleeping Beauties with all-Heads or all-Tails showing and a new Beauty must be anesthetized with a new coin; the queue length changes over time and sometimes switches face. You can stop the experiment when the queue is empty, as a random walk guarantees to happen eventually, if you like tying up loose ends.