Given that Beauty is being asked the question, the probability that heads had come up is 1⁄3. This doesn’t mean the probability of heads itself is 1⁄3. So I think this is a confusion about what the question is asking. Is the question asking what is the probability of heads, or what is the probability of heads given an awakening?
Bayes theorem:
x = # of times awakened after heads
y = # of times awakened after tails
p(heads/awakened) = n(heads and awakened) / n(awakened) = x / (x+y)
Yields 1⁄3 when x=1 and y=2.
Where is the probability of heads? Actually we already assumed in the calculation above that p(heads) = 0.5. For a general biased coin, the calculation is slightly more complex:
I’m leaving this comment because I think the equations help explain how the probability-of-heads and the probability-of-heads-given-awakening are inter-related but, obviously—I know you know this already—not the same thing.
To clarify, since the probability-of-heads and the probability-of-heads-given-single-awakening-event are different things, it is indeed a matter of semantics: if Beauty is asked about the probability of heads per event … what is the event? Is the event the flip of the coin (p=1/2) or an awakening (p=1/3)? In the post narrative, this remains unclear.
Which event is meant would become clear if it was a wager (and, generally, if anything whatsoever rested on the question). For example: if she is paid per coin flip for being correct (event=coin flip) then she should bet heads to be correct 1 out of 2 times; if she is paid per awakening for being correct (event=awakening) then she should bet tails to be correct 2 out of 3 times.
Actually .. arguing with myself now .. Beauty wasn’t asked about a probability, she was asked if she thought heads had been flipped, in the past. So this is clear after all—did she think heads was flipped, or not?
Viewing it this way, I see the isomorphism with the class of anthropic arguments that ask if you can deduce something about the longevity of humans given that you are an early human. (Being a human in a certain century is like awakening on a certain day.) I suppose then my solution should be the same. Waking up is not evidence either way that heads or tails was flipped. Since her subjective experience is the same however the coin is flipped (she wakes up) she cannot update upon awakening that it is more likely that tails was flipped. Not even if flipping tails means she wakes up 10 billion times more than if heads was flipped.
However, I will think longer if there are any significant differences between the two problems. Thoughts?
Why was this comment down-voted so low? (I rarely ask, but this time I can’t guess.) Is it too basic math? If people are going to argue whether 1⁄3 or 1⁄2, I think it is useful to know their debating about two different probabilities: the probability of heads or the probability of heads given an awakening.
By “awakened” here you mean “awakened at all”. I think you’ve shown already that the probability that heads was flipped given that she was awakened at all is 1⁄2, since in both cases she’s awakened at all and the probability of heads is 1⁄2. I think your dispute is with people who don’t think “I was awakened at all” is all that Beauty knows when she wakes up.
Beauty also knows how many times she it likely to have been woken up when the coin lands heads—and the same for tails. She knew that from the start of the experiment.
OK, I see now why you are emphasizing being awoken at all. That is the relevant event, because that is exactly what she experiences and all that she has to base her decision upon.
(But keep in mind that people are just busy answering different questions, they’re not necessarily incorrect for answering a different question.)
Given that Beauty is being asked the question, the probability that heads had come up is 1⁄3. This doesn’t mean the probability of heads itself is 1⁄3. So I think this is a confusion about what the question is asking. Is the question asking what is the probability of heads, or what is the probability of heads given an awakening?
Bayes theorem:
x = # of times awakened after heads
y = # of times awakened after tails
p(heads/awakened) = n(heads and awakened) / n(awakened) = x / (x+y)
Yields 1⁄3 when x=1 and y=2.
Where is the probability of heads? Actually we already assumed in the calculation above that p(heads) = 0.5. For a general biased coin, the calculation is slightly more complex:
p(H) =probability of heads
p(T) = probability of tails
x = # of times awakened after heads
y = # of times awakened after tails
p(heads/awakened) = n(heads and awakened) / n(awakened) = p(H)x / (p(H)x + p(T)y)
Yields 1⁄3 when x=1 and y=2 and p(H)=p(T)=0.5.
I’m leaving this comment because I think the equations help explain how the probability-of-heads and the probability-of-heads-given-awakening are inter-related but, obviously—I know you know this already—not the same thing.
To clarify, since the probability-of-heads and the probability-of-heads-given-single-awakening-event are different things, it is indeed a matter of semantics: if Beauty is asked about the probability of heads per event … what is the event? Is the event the flip of the coin (p=1/2) or an awakening (p=1/3)? In the post narrative, this remains unclear.
Which event is meant would become clear if it was a wager (and, generally, if anything whatsoever rested on the question). For example: if she is paid per coin flip for being correct (event=coin flip) then she should bet heads to be correct 1 out of 2 times; if she is paid per awakening for being correct (event=awakening) then she should bet tails to be correct 2 out of 3 times.
Actually .. arguing with myself now .. Beauty wasn’t asked about a probability, she was asked if she thought heads had been flipped, in the past. So this is clear after all—did she think heads was flipped, or not?
Viewing it this way, I see the isomorphism with the class of anthropic arguments that ask if you can deduce something about the longevity of humans given that you are an early human. (Being a human in a certain century is like awakening on a certain day.) I suppose then my solution should be the same. Waking up is not evidence either way that heads or tails was flipped. Since her subjective experience is the same however the coin is flipped (she wakes up) she cannot update upon awakening that it is more likely that tails was flipped. Not even if flipping tails means she wakes up 10 billion times more than if heads was flipped.
However, I will think longer if there are any significant differences between the two problems. Thoughts?
Why was this comment down-voted so low? (I rarely ask, but this time I can’t guess.) Is it too basic math? If people are going to argue whether 1⁄3 or 1⁄2, I think it is useful to know their debating about two different probabilities: the probability of heads or the probability of heads given an awakening.
This is incorrect.
Given that Beauty is being asked the question, the probability that heads had come up is 1⁄2.
This is bayes’ theorem:
p(H)=1/2
p(awakened|H)=p(awakened|T)=1
P(H|awakened)=p(awakened|H)P(H)/(p(awakened|H)p(H)+p(awakened|T)p(T))
which equals 1⁄2
By “awakened” here you mean “awakened at all”. I think you’ve shown already that the probability that heads was flipped given that she was awakened at all is 1⁄2, since in both cases she’s awakened at all and the probability of heads is 1⁄2. I think your dispute is with people who don’t think “I was awakened at all” is all that Beauty knows when she wakes up.
Beauty also knows how many times she it likely to have been woken up when the coin lands heads—and the same for tails. She knew that from the start of the experiment.
OK, I see now why you are emphasizing being awoken at all. That is the relevant event, because that is exactly what she experiences and all that she has to base her decision upon.
(But keep in mind that people are just busy answering different questions, they’re not necessarily incorrect for answering a different question.)