What she is really surprised about however, is not that she has observed an unlikely event ({HHTHTHHT}), but that she has observed an unexpected pattern.
Why do you oppose these two things to each other? Talking about patterns is just another way to describe the same fact.
In this case, the coincidence of the sequence she had in mind and the sequence produced by the coin tosses constitutes a symmetry which our mind readily detects and classifies as such a pattern.
Well, yes. Or you can say that having a specific combination in mind allowed to observe event “this specific combination” instead of “any combination”. Once again this is just using different language to talk about the same thing.
One could also say that she has not just observed the event {HHTHTHHT} alone, but also the coincidence which can be regarded as an event, too. Both events, the actual coin toss sequence and the coincidence, are unlikely events and both become extremely unlikely with longer sequences.
Oh! Are you saying that she has observed the intersection of two rare events: “HHTHTHHT was produced by coin tossing” and “HHTHTHHT was the sequence that I came up with in my mind” both of which have probability 1/2^8 so now she is surprised as if she observed an event with (1/2^8)^2?
That’s not actually the case. If the person came up with some other combination and then it was realized on the coin tosses the surprise would be the same—there are 1/2^8 degrees of dreedom here—for every possible combination of Heads and Tails with lenghth 8. So the probability of the observed event is still 1/2^8.
Maybe I expressed myself somewhat misleadingly. I am not saying that she is surprised because the coincidence is more unlikely than the sequence. You are absolutely right in correcting me that the latter isn‘t even the case (also since P(HHTHTHHT/„HHTHTHHT“)=P(HHTHTHHT)=1/2^8). What I was trying to say is that her suprise about the coincidence arises from the circumtance that the coincidence is both unlikely and looks like a pattern. That fact that an event is unlikely is a necessary condition to be surprised about its occurence but not a sufficient condition.
I agree with you when you are saying that how we structure our perception of the world is biased in some way towards what we are „tracking“ in our minds. And I also agree that this bias could be mathematically modelled by the event spaces you are proposing. But I would not go too far to say that we do only observe such events we are currently tracking (please let me know if I misread you or you feel strawmanned here, since it is absolutely not my intention to annoy you!). If this was true, then we could not observe the event „any other coin sequence“ as well since this event is by definition not being tracked. In fact, in order to detect a correspondence between a coin sequence that we have in mind and the actual sequence, our brain has to compare them to decide if there is a match. I can hardly imagine how this comparison could work without observing the specific actual sequence in the first place. That we classify and perceive a specific sequence as „any other sequence“ can be the result of the comparison, but is not its starting point.
In conclusion, I do not see a contradiction in not being surprised to observe an extremely unlikely event.
Why do you oppose these two things to each other? Talking about patterns is just another way to describe the same fact.
Well, yes. Or you can say that having a specific combination in mind allowed to observe event “this specific combination” instead of “any combination”. Once again this is just using different language to talk about the same thing.
Oh! Are you saying that she has observed the intersection of two rare events: “HHTHTHHT was produced by coin tossing” and “HHTHTHHT was the sequence that I came up with in my mind” both of which have probability 1/2^8 so now she is surprised as if she observed an event with (1/2^8)^2?
That’s not actually the case. If the person came up with some other combination and then it was realized on the coin tosses the surprise would be the same—there are 1/2^8 degrees of dreedom here—for every possible combination of Heads and Tails with lenghth 8. So the probability of the observed event is still 1/2^8.
Maybe I expressed myself somewhat misleadingly. I am not saying that she is surprised because the coincidence is more unlikely than the sequence. You are absolutely right in correcting me that the latter isn‘t even the case (also since P(HHTHTHHT/„HHTHTHHT“)=P(HHTHTHHT)=1/2^8). What I was trying to say is that her suprise about the coincidence arises from the circumtance that the coincidence is both unlikely and looks like a pattern. That fact that an event is unlikely is a necessary condition to be surprised about its occurence but not a sufficient condition.
I agree with you when you are saying that how we structure our perception of the world is biased in some way towards what we are „tracking“ in our minds. And I also agree that this bias could be mathematically modelled by the event spaces you are proposing. But I would not go too far to say that we do only observe such events we are currently tracking (please let me know if I misread you or you feel strawmanned here, since it is absolutely not my intention to annoy you!). If this was true, then we could not observe the event „any other coin sequence“ as well since this event is by definition not being tracked. In fact, in order to detect a correspondence between a coin sequence that we have in mind and the actual sequence, our brain has to compare them to decide if there is a match. I can hardly imagine how this comparison could work without observing the specific actual sequence in the first place. That we classify and perceive a specific sequence as „any other sequence“ can be the result of the comparison, but is not its starting point.
In conclusion, I do not see a contradiction in not being surprised to observe an extremely unlikely event.