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 tails
Before being told anything, you should estimate a 2/3 chance that it's Monday (not a 75% chance). There are three possibilities: heads/Monday, tails/Monday, and tails/Tuesday, all of which are equally likely. Because tails results in two awakenings, and you are calculating probability per awakening, that boosts the probability of tails, so it would be incorrect to put 50% on heads/Monday and 25% on tails/Monday. Tails/Monday is not half as likely as heads/Monday; it is equally likely. Only in the scenario where you were woken up either on Monday or Tuesday, but not both, would the probability of tails/Monday be 25%.
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.
When you are told that it is Monday, the chance is not 50/75, it's (1/3) / (2/3) = 50%. Being told that it is Monday does increase the probability that the result is heads; however, it increases it from 1/3 -> 1/2, not from 1/2 -> 2/3.
Before being told anything, you should estimate a 2/3 chance that it's Monday (not a 75% chance). There are three possibilities: heads/Monday, tails/Monday, and tails/Tuesday, all of which are equally likely.
I disagree that these situations are equally likely. We can understand it better by taking the extreme example. I will be much more surprised to hear that the coin was tails and that we are now at day #500,000, then that the coin was heads and that it is the first day. So obviously these two situations do not seem equally likely to me. And in partic...
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