I misplaced that comment. It was not a response to yours.
More likely than what?
More likely than .5. In fact I am saying the probability of getting heads is 2/3 after being told that it is Monday.
Using per-awakening probabilities, ithe probability of heads without this information is 1/3.
This is a frequentist definition of probability. I am using probability as a subjective degree of belief, where being almost certain that something is so means assigning a probability near 1, being almost certain that it is not means assigning a probability near 0, and being completely unsure means .5.
Here is how this works. If I am sleeping Beauty, on every awakening I am subjectively in the same condition. I am completely unsure whether the coin landed/will land heads or tails. So the probability of heads is .5, and the probability of tails is .5.
What is the subjective probability that it is Monday, and what is the subjective probability it is Tuesday? It is easier to understand if you consider the extreme form. Let's say that if the coin lands tails, I will be woken up 1,000,000 times. I will be quite surprised if I am told that it is day #500,000, or any other easily definable number. So my degree of belief that it is day #500,000 has to be quite low. On the other hand, if I am told that it is the first day, that will be quite unsurprising. But it will be unsurprising mainly because there is a 50% chance that will be the only awakening anyway. This tells me that before I am told what day it is, my estimate of the probability that it is the first day is a tiny bit more than 50% -- 50% of this is from the possibility that the coin landed heads, and a tiny bit more from the possibility that it landed tails but it is still the first day.
When we transition to the non-extreme form, being Monday is still less surprising than being Tuesday. In fact, before being told anything, I estimate a chance of 75% that it is Monday -- 50% from the coin landing heads, and another 25% from the coin landing tails. And when I am told that it is in fact Monday, then I think there is a chance of 2/3, i.e. 50/75, that the coin will land heads.
This tells me that before I am told what day it is, my estimate of the probability that it is the first day is a tiny bit more than 50%... When we transition to the non-extreme form, being Monday is still less surprising than being Tuesday.
In the non-extreme form, the chance of being Monday is 2/3 and the chance of being Tuesday is 1/3. 2/3 is indeed less surprising than 1/3, so your reasoning is correct.
...before being told anything, I estimate a chance of 75% that it is Monday -- 50% from the coin landing heads, and another 25% from the coin landing
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