You flip a fair coin 20 times. If this sequence contains at least one HHHH, I pay you $100. If it contains at least one HHHT, you pay me $100. If it contains neither, nobody wins.
Nits could be picked about this working more because “occurrences of a given substring matched continuously within a longer string” is a different question from “odds of a given string”; rather than because of irregular strings being inherently more probable or any difference between finite and infinite strings.
Specifically the part where, for the HHHT player, if the string is at HHH, then either they get a successful match from a T on the next flip or the string stands at HHHH and they can still hope for H[HHHT] a mere one flip later (compared to the HHHH player having to start over from zero whenever a T comes up). Benefiting greatly there from the target string overlapping with itself as you slide a 4-wide frame along the larger sequence.
The imbalance thus created would presumably still appear if you were to count matches on a similar sliding basis along an infinite string. Or equally disappear in the finite case if you only look at discrete chunks of 4 flips at a time (and treat that 20 flip sequence as 5 independent nonoverlapping trials).
So the claim would have to be that the bias is adaptive because we’re more likely to need to intuitively estimate odds about occurrences in continuous series rather than discrete chunks. Which isn’t implausible, but is less intrinsically obvious than the idea that we’d more often encounter finite cases than infinite ones.
Why are you calling this a nitpick? IMO it’s a major problem with the post—I was very unhappy that no mention was made of this obvious problem with the reasoning presented.
Because the central idea of the post isn’t really about that specific probability puzzle, and can in theory stand alone to succeed or fail on other merits—regardless of whether that illustrative example in particular is actually a good choice of example.
Possibly there are better examples in the full paper linked, but I couldn’t comment on that either way because I’ve only read this excerpt/summary.
Nits could be picked about this working more because “occurrences of a given substring matched continuously within a longer string” is a different question from “odds of a given string”; rather than because of irregular strings being inherently more probable or any difference between finite and infinite strings.
Specifically the part where, for the HHHT player, if the string is at HHH, then either they get a successful match from a T on the next flip or the string stands at HHHH and they can still hope for H[HHHT] a mere one flip later (compared to the HHHH player having to start over from zero whenever a T comes up). Benefiting greatly there from the target string overlapping with itself as you slide a 4-wide frame along the larger sequence.
The imbalance thus created would presumably still appear if you were to count matches on a similar sliding basis along an infinite string. Or equally disappear in the finite case if you only look at discrete chunks of 4 flips at a time (and treat that 20 flip sequence as 5 independent nonoverlapping trials).
So the claim would have to be that the bias is adaptive because we’re more likely to need to intuitively estimate odds about occurrences in continuous series rather than discrete chunks. Which isn’t implausible, but is less intrinsically obvious than the idea that we’d more often encounter finite cases than infinite ones.
Why are you calling this a nitpick? IMO it’s a major problem with the post—I was very unhappy that no mention was made of this obvious problem with the reasoning presented.
Because the central idea of the post isn’t really about that specific probability puzzle, and can in theory stand alone to succeed or fail on other merits—regardless of whether that illustrative example in particular is actually a good choice of example.
Possibly there are better examples in the full paper linked, but I couldn’t comment on that either way because I’ve only read this excerpt/summary.