First of all, that can’t possibly be right.

I understand that it all may be somewhat counterintuitive. I’ll try to answer whatever questions you have. If you think you have some way to formally define what “Today” means in Sleeping Beauty—feel free to try.

Second of all, it goes against everything you’ve been saying for the entire series.

Seems very much in accordance with what I’ve been saying.

Throughout the series I keep repeating the point that all we need to solve anthropics is to follow probability theory where it leads and then there will be no paradoxes. This is exactly what I’m doing here. There is no formal way to define “Today is Monday” in Sleeping Beauty and so I simply accept this, as the math tells me to, and then the “paradox” immediately resolves.

Suppose someone who has never heard of the experiment happens to call sleeping beauty on her cell phone during the experiment and ask her “hey, my watch died and now I don’t know what day it is; could you tell me whether today is Monday or Tuesday?” (This is probably a breach of protocol and they should have confiscated her phone until the end, but let’s ignore that.).

Are you saying that she has no good way to reason mathematically about that question? Suppose they told her “I’ll pay you a hundred bucks if it turns out you’re right, and it costs you nothing to be wrong, please just give me your best guess”. Are you saying there’s no way for her to make a good guess? If you’re not saying that, then since probabilities are more basic than utilities, shouldn’t she also have a credence?

Good question. First of all, as we are talking about betting I recommend you read the next post, where I explore it in more details, especially if you are not fluent in expected utility calculations.

Secondly, we can’t ignore the breach of the protocol. You see, if anything breaks the symmetry between awakening, the experiment changes in a substantial manner. See Rare Event Sleeping Beauty, where probability that the coin is Heads can actually be ^{1}⁄_{3}.

But we can make a similar situation without breaking the symmetry. Suppose that on every awakening a researcher comes to the room and proposes the Beauty to bet on which day it currently is. At which odds should the Beauty take the bet?

This is essentially the same betting scheme as ice-cream stand, which I deal with in the end of the previous comment.

Exactly! I’m glad that you actually engaged with the problem.

The first step is to realize that here “today” can’t mean “Monday xor Tuesday” because such event never happens. On every iteration of experiment both Monday and Tuesday are realized. So we can’t say that the participant knows that they are awakened on Monday xor Tuesday.

Can we say that participant knows that they are awakened on Monday or Tuesday? Sure. As a matter of fact:

P(Monday or Tuesday) = 1

P(Heads|Monday or Tuesday) = P(Heads) =

^{1}⁄_{2}This works, here probability that the coin is Heads in this iteration of the experiment happens to be the same as what our intuition is telling us P(Heads|Today) is supposed to be, however we still can’t define “Today is Monday”:

P(Monday|Monday or Tuesday) = P(Monday) = 1

Which doesn’t fit our intuition.

How can this be? How can we have a seeminglly well-defined probability for “Today the coin is Heads” but not for “Today is Monday”? Either “Today” is well-defined or it’s not, right? Take some time to think about it.

What do we

actually meanwhen we say that on an awakening the participant supposed to believe that the coin is Heads with 50% probability? Is it really aboutthis day in particular? Or is it about something else?The answer is: we actually mean, that on

any day of the experimentbeit Monday or Tuesdaythe participant is supposed to believe that the coin is Heads with 50% probability. We can not formally specify “Today” in this problem but there is a clever, almost cheating way to specify “Anyday” without breaking anything.This is not easy. It requires a way to define P(A|B), when P(B) itself is undefined which is unconventional. But, moreover, it requires symmetry. P(Heads|Monday) has to be equal to P(Heads|Tuesday) only then we have a coherent P(Heads|Anyday).