Your update doesn’t solve the problem. It’s a semantic issue about what credence we are being asked. If we are being asked about the probability of our coin flip associated with this iteration of the experiment, then the answer is 1⁄2. If we are being asked about the probability of the coin flip associated with this particular awakening, then it must be 1⁄3.
You say that you must use cell counts of 500,250,250, but the fact is that if you repeat the experiment 1000 times, sleeping beauty will be awoken 1500 times, not 1000. So what are you doing with the other 500 awakenings? I would say you are implicitly ignoring them, as you do when you say “we only accept her last decision” in the bet scenario. The reformulations of this using different people, rather than the same person being awoken multiple times don’t seem to cause as much trouble.
The semantic issue here is reminiscent of arguments I’ve seen over the Monty Hall problem when it is misstated so that Monty’s algorithm is not clear. People who assume what he usually does on the show come up with 2⁄3, and people who don’t make any assumptions come up with 1⁄2 (as do most of the people who simply don’t understand restricted choice).
Re: “If we are being asked about the probability of our coin flip associated with this iteration of the experiment, then the answer is 1⁄2. If we are being asked about the probability of the coin flip associated with this particular awakening, then it must be 1⁄3.”
What is actually asked at each awakening is:
“What is your credence now for the proposition that our coin landed heads?”
I figure that makes the answer 1⁄3 - and not 1⁄2.
If the question had been: “What is your credence that this is the last time you awaken and our coin landed heads-up?”
Your update doesn’t solve the problem. It’s a semantic issue about what credence we are being asked. If we are being asked about the probability of our coin flip associated with this iteration of the experiment, then the answer is 1⁄2. If we are being asked about the probability of the coin flip associated with this particular awakening, then it must be 1⁄3.
You say that you must use cell counts of 500,250,250, but the fact is that if you repeat the experiment 1000 times, sleeping beauty will be awoken 1500 times, not 1000. So what are you doing with the other 500 awakenings? I would say you are implicitly ignoring them, as you do when you say “we only accept her last decision” in the bet scenario. The reformulations of this using different people, rather than the same person being awoken multiple times don’t seem to cause as much trouble.
The semantic issue here is reminiscent of arguments I’ve seen over the Monty Hall problem when it is misstated so that Monty’s algorithm is not clear. People who assume what he usually does on the show come up with 2⁄3, and people who don’t make any assumptions come up with 1⁄2 (as do most of the people who simply don’t understand restricted choice).
Re: “If we are being asked about the probability of our coin flip associated with this iteration of the experiment, then the answer is 1⁄2. If we are being asked about the probability of the coin flip associated with this particular awakening, then it must be 1⁄3.”
What is actually asked at each awakening is:
“What is your credence now for the proposition that our coin landed heads?”
I figure that makes the answer 1⁄3 - and not 1⁄2.
If the question had been: “What is your credence that this is the last time you awaken and our coin landed heads-up?”
...then the answer would have been 1⁄2.
...but that wasn’t the question that was asked.
Cool story bro