A good way to resolve the paradox to me is to modify the code to combine both the functions into one function and record the sequences of the 10,000, In one array you store the sequences where there are two consecutive 6′s and in the second you store the one where they are not consecutive. That makes it a bit clearer.
For a run of 10,000 I get 412 runs where the first two 6′s are consecutive (sequences_no_gap), and 192 where they are not (sequences_can_gap). So if its just case A you get 412 runs, but for case B you get 412+192 runs. Then you look at the average sequence length of sequences_no_gap and compare it to sequences_can_gap. If the average sequence length in sequences_can_gap > than sequences_no_gap, then that means the expectation will be higher, and thats what you get.
I did not believe this until I tried it myself, so yes, very paradox, much confusing.
A good way to resolve the paradox to me is to modify the code to combine both the functions into one function and record the sequences of the 10,000, In one array you store the sequences where there are two consecutive 6′s and in the second you store the one where they are not consecutive. That makes it a bit clearer.
For a run of 10,000 I get 412 runs where the first two 6′s are consecutive (sequences_no_gap), and 192 where they are not (sequences_can_gap). So if its just case A you get 412 runs, but for case B you get 412+192 runs. Then you look at the average sequence length of sequences_no_gap and compare it to sequences_can_gap. If the average sequence length in sequences_can_gap > than sequences_no_gap, then that means the expectation will be higher, and thats what you get.
mean sequence lengths
sequences_can_gap: 3.93
sequences_no_gap: 2.49
Examples:
sequences_no_gap
[[4, 6, 6], [6, 6], [6, 6], [4, 6, 6], [6, 6], [6, 6], [6, 6], [6, 6], [6, 6], [6, 6], [6, 6], [6, 6], [4, 6, 6], [4, 6, 6], …]
sequences_can_gap
[[6, 4, 4, 6], [6, 4, 6], [4, 6, 4, 6], [2, 2, 6, 2, 6], [6, 4, 2, 6], [6, 4, 6], [6, 2, 6], [6, 2, 4, 6], [6, 2, 2, 4, 6], [6, 4, 6], [6, 4, 6], [2, 4, 6, 2, 6], [6, 4, 6], [6, 4, 6], …]
The many examples such as [6 4 4 6] which are excluded in the first case make the expected number of rolls higher for the case where they are allowed.
(Note GPT o-1 is confused by this problem and gives slop)