I’m a halfer, but think you did your math wrong when calculating the thirder view.
The thirder view is that the probability of an event happening is the experimenter’s expectation of the proportion of awakenings where the event happened.
So for your setup, with k=2:
There are three possible outcomes: H, HT, and TT.
H happens in 50% of experiments, HT happens in 25% and TT happens in 25%.
When H happens there is 1 awakening, when HT happens there are 2 awakenings, and when TT happens there are 4 awakenings.
We’ll imagine that the experiment is run 4 times, and that H happened in 2 of them, HT happened once, and TT happened once. This results in 2*1=2 H awakenings, 1*2=2 HT awakenings, and 1*4=4 TT awakenings.
Therefore, H happens in 2/(2+2+4)=25% of awakenings, HT happens in 25% of awakenings, and TT happens in 50% of awakenings.
The thirder view is thus that upon awakening Beauty’s credence that the coin came up heads should be 25%.
What is you [sic] credence that in this experiment the coin was tossed k times and the outcome of the k-th toss is Tails?
Answering your question, the thirder view is that there was a 6/8=75% chance the coin was tossed twice, and a 4⁄6 chance that the second toss was a tails conditional on it being the case that two tosses were made.
Unconditionally, the thirder’s credence is 4/8=50% chance that it is both true that the coin was tossed two times and that the second toss was a tails.
It seems that I’ve been sloppy and therefore indeed misrepresented thirders reasoning here. Shame on me. Will keep this post available till tomorrow, as a punishment for myself and then back to the drawing board.
I’m a halfer, but think you did your math wrong when calculating the thirder view.
The thirder view is that the probability of an event happening is the experimenter’s expectation of the proportion of awakenings where the event happened.
So for your setup, with k=2:
There are three possible outcomes: H, HT, and TT.
H happens in 50% of experiments, HT happens in 25% and TT happens in 25%.
When H happens there is 1 awakening, when HT happens there are 2 awakenings, and when TT happens there are 4 awakenings.
We’ll imagine that the experiment is run 4 times, and that H happened in 2 of them, HT happened once, and TT happened once. This results in 2*1=2 H awakenings, 1*2=2 HT awakenings, and 1*4=4 TT awakenings.
Therefore, H happens in 2/(2+2+4)=25% of awakenings, HT happens in 25% of awakenings, and TT happens in 50% of awakenings.
The thirder view is thus that upon awakening Beauty’s credence that the coin came up heads should be 25%.
Answering your question, the thirder view is that there was a 6/8=75% chance the coin was tossed twice, and a 4⁄6 chance that the second toss was a tails conditional on it being the case that two tosses were made.
Unconditionally, the thirder’s credence is 4/8=50% chance that it is both true that the coin was tossed two times and that the second toss was a tails.
Thank you!
It seems that I’ve been sloppy and therefore indeed misrepresented thirders reasoning here. Shame on me. Will keep this post available till tomorrow, as a punishment for myself and then back to the drawing board.