Yes, I also had this concern when I read that paragraph.
If Sleeping Beauty were paid $1 for a correct guess as to which day it is, she should probably always answer “today is Monday”.
If the coin came up heads, she is correct. If the coin came up tails, she will be correct on Monday and wrong on Tuesday. So this guarantees that she earns $1 every time the experiment runs.
If she instead always answers “today is Tuesday”, then if the coin came up heads, she earns nothing; if the coin came up tails, she earns nothing on Monday and $1 on Tuesday. So this strategy has an expected revenue of $0.50.
So is it fair to “say that Monday is consistent with observations, as is Tuesday, and refuse to speculate further”? I think that’s yielding too much ground.
This particular case isn’t a philosophical question about getting information from knowledge of being an observer among all sets of observers. It’s a purely pragmatic analysis that says you’ll always be woken up on a Monday but not always on a Tuesday, so Monday awakenings are simply more probable. That’s the structure of the experiment, and has nothing to do with the anthropic principle.
An analogous experiment that removes the observer is: The experimenter has a bag with only a white ball and a green ball in it. He draws once from the bag randomly, but if he drew the white ball, he draws again. Anyone can tell you that he’ll produce a green ball every round, but a white ball only half the time. If it’s done behind a screen and you’re asked “Was a green ball drawn this round?”, the smart bet is yes.
Yes, I also had this concern when I read that paragraph.
If Sleeping Beauty were paid $1 for a correct guess as to which day it is, she should probably always answer “today is Monday”.
If the coin came up heads, she is correct. If the coin came up tails, she will be correct on Monday and wrong on Tuesday. So this guarantees that she earns $1 every time the experiment runs.
If she instead always answers “today is Tuesday”, then if the coin came up heads, she earns nothing; if the coin came up tails, she earns nothing on Monday and $1 on Tuesday. So this strategy has an expected revenue of $0.50.
So is it fair to “say that Monday is consistent with observations, as is Tuesday, and refuse to speculate further”? I think that’s yielding too much ground.
This particular case isn’t a philosophical question about getting information from knowledge of being an observer among all sets of observers. It’s a purely pragmatic analysis that says you’ll always be woken up on a Monday but not always on a Tuesday, so Monday awakenings are simply more probable. That’s the structure of the experiment, and has nothing to do with the anthropic principle.
An analogous experiment that removes the observer is: The experimenter has a bag with only a white ball and a green ball in it. He draws once from the bag randomly, but if he drew the white ball, he draws again. Anyone can tell you that he’ll produce a green ball every round, but a white ball only half the time. If it’s done behind a screen and you’re asked “Was a green ball drawn this round?”, the smart bet is yes.