Echoing Said’s comment, what does it mean to be “correct” in this context? If we ask Beauty to pick between heads or tails, and she picks heads, then sometimes this will be correct, and sometimes not.
In order for Beauty to give a (correct) probabilistic answer to the question (1/3 or 1⁄2) we need to introduce the idea of some proportion of trials. We need to at least imagine running the situation many times, and talk about some proportion of those imagined repeats. These imagined trials don’t need to actually happen, they are imaginary. But they are an indispensable fiction.
Now, we imagine 100 repeats. 50 heads, 50 tails. Beauty is awoken a total 150 times. For 100 awakenings it was a head that was flicked, for 50 awakenings a tail.
>For 1/3rd of the awakenings the coin was tails. For 1⁄2 of the trials the coin was tails.
I don’t think anyone (halfer or thirder) disputes the line directly above (with the >). There is agreement on what proportion of awakenings tails was tossed, and on what proportion of trials a tails was tossed. We can all see that one of the two proportions is 1⁄3 and the other is 1⁄2. Which of the two proportions is picked out by the word “probability” is the entire argument.
The rewards structure @Said Achmiz is talking about is a nice way of making people either aim to be right in as many guesses as possible or in as many trials as possible, which demand different strategies.
Echoing Said’s comment, what does it mean to be “correct” in this context? If we ask Beauty to pick between heads or tails, and she picks heads, then sometimes this will be correct, and sometimes not.
In order for Beauty to give a (correct) probabilistic answer to the question (1/3 or 1⁄2) we need to introduce the idea of some proportion of trials. We need to at least imagine running the situation many times, and talk about some proportion of those imagined repeats. These imagined trials don’t need to actually happen, they are imaginary. But they are an indispensable fiction.
Now, we imagine 100 repeats. 50 heads, 50 tails. Beauty is awoken a total 150 times. For 100 awakenings it was a head that was flicked, for 50 awakenings a tail.
>For 1/3rd of the awakenings the coin was tails. For 1⁄2 of the trials the coin was tails.
I don’t think anyone (halfer or thirder) disputes the line directly above (with the >). There is agreement on what proportion of awakenings tails was tossed, and on what proportion of trials a tails was tossed. We can all see that one of the two proportions is 1⁄3 and the other is 1⁄2. Which of the two proportions is picked out by the word “probability” is the entire argument.
The rewards structure @Said Achmiz is talking about is a nice way of making people either aim to be right in as many guesses as possible or in as many trials as possible, which demand different strategies.