Yes, rather than resolving the surprise of “the exact sequence HHTHTTHTTH” by declaring that it shouldn’t be part of the set of events, I would prefer to resolve it via something like:
It should be part of the set of events I’m allowed to consider just like any other subset of all 10-flip sequences.
We do observe events (or outcomes that if constructed as singleton events) all the time that would we would have predicted to be exceedingly improbable (while they may be improbable individually, a union of them may not be).
Observing some particular unlikely event like “the exact sequence HHTHTTHTTH occurs” should in fact raise my relative belief in any hypothesis by a large factor if that hypothesis would have uniquely predicted that to occur, as compared to others that would have made a far more non-specific prediction. (up to a factor of at most 2^10 unless the other hypothesis considered that sequence to be unlikelier than uniform)
Even if all this is true, I still do not and should not feel surprised in such a case because I think surprise has more to do the amount by which something shifts the beliefs I have that my brain intuits to be important for various reasons. It has little to do with the likelihood of events I observe, other than how it affects those beliefs. I didn’t have any prior reason to assign any meaningful weight to hypotheses about the coin that would predict that exact sequence and no others, such that even after scaling them by a large factor, my overall beliefs about the coin and the distribution of likely future flips should remain very similar to before, therefore I feel little surprise.
By contrast I might feel a little more surprise seeing “HHHHHHHHHH”. And again the reason is not really because of the likelihood or unlikelihood of that sequence, and it also has little to do with which sequences I’m being told I can define to be a mathematical event or not. Rather I think it’s closer to something like “this coin is biased heads” or “this coin always flips heads” are competing hypotheses to “this coin is fair” that while initially extremely unlikely would not be outlandish to consider, and if true it would affect my conception of the coin and predictions of its future flips. So this time the large relative boost would come closer to shifting my beliefs in a way that would impact how I think about the coin and make future predictions, therefore I feel more surprise.
Yes, rather than resolving the surprise of “the exact sequence HHTHTTHTTH” by declaring that it shouldn’t be part of the set of events, I would prefer to resolve it via something like:
It should be part of the set of events I’m allowed to consider just like any other subset of all 10-flip sequences.
We do observe events (or outcomes that if constructed as singleton events) all the time that would we would have predicted to be exceedingly improbable (while they may be improbable individually, a union of them may not be).
Observing some particular unlikely event like “the exact sequence HHTHTTHTTH occurs” should in fact raise my relative belief in any hypothesis by a large factor if that hypothesis would have uniquely predicted that to occur, as compared to others that would have made a far more non-specific prediction. (up to a factor of at most 2^10 unless the other hypothesis considered that sequence to be unlikelier than uniform)
Even if all this is true, I still do not and should not feel surprised in such a case because I think surprise has more to do the amount by which something shifts the beliefs I have that my brain intuits to be important for various reasons. It has little to do with the likelihood of events I observe, other than how it affects those beliefs. I didn’t have any prior reason to assign any meaningful weight to hypotheses about the coin that would predict that exact sequence and no others, such that even after scaling them by a large factor, my overall beliefs about the coin and the distribution of likely future flips should remain very similar to before, therefore I feel little surprise.
By contrast I might feel a little more surprise seeing “HHHHHHHHHH”. And again the reason is not really because of the likelihood or unlikelihood of that sequence, and it also has little to do with which sequences I’m being told I can define to be a mathematical event or not. Rather I think it’s closer to something like “this coin is biased heads” or “this coin always flips heads” are competing hypotheses to “this coin is fair” that while initially extremely unlikely would not be outlandish to consider, and if true it would affect my conception of the coin and predictions of its future flips. So this time the large relative boost would come closer to shifting my beliefs in a way that would impact how I think about the coin and make future predictions, therefore I feel more surprise.