HHHT does take less time to show up than HHHH in repeated simulations, and is more commonly encountered in small samples.
This is true in specific technical ways and false in specific technical ways. It’s far from obvious to me that the true ways are more important than the false ways. Here are some ways we can cash this out, along with what I think are the results:
Continue flipping until we generate either sequence, then stop. Which did we most likely encounter? Equally likely.
Continue flipping until we generate a specific sequence. What’s the expected stopping time? Lower for HHHT.
Generate a sample of size > 5. How many of each sequence does it have? More HHHH.
Generate a sample of size > 5. How likely is each sequence to show up at least once? HHHT is more likely. This is the bet.
Even accepting the claim as basically true, it’s because the end overlaps the beginning, not because of regularity. This is a type of regularity, I admit, but I don’t believe it’s well correlated with what people will perceive as statistical regularity. I think you get the same results with HTH versus HTT (replacing HHHH and HHHT respectively).
I would expect a less pronounced version of the same effect. Both get to HT together, but if you’re looking for HTH and get HTT then you’re starting over hoping for your first H on the next flip, whereas if you’re looking for HTT and get HTH you’ve got a small headstart because that last H can be the first H going forward to HT[HTT]
Somewhat duplicating noggin-scratcher’s comment.
This is true in specific technical ways and false in specific technical ways. It’s far from obvious to me that the true ways are more important than the false ways. Here are some ways we can cash this out, along with what I think are the results:
Continue flipping until we generate either sequence, then stop. Which did we most likely encounter? Equally likely.
Continue flipping until we generate a specific sequence. What’s the expected stopping time? Lower for HHHT.
Generate a sample of size > 5. How many of each sequence does it have? More HHHH.
Generate a sample of size > 5. How likely is each sequence to show up at least once? HHHT is more likely. This is the bet.
Even accepting the claim as basically true, it’s because the end overlaps the beginning, not because of regularity. This is a type of regularity, I admit, but I don’t believe it’s well correlated with what people will perceive as statistical regularity. I think you get the same results with HTH versus HTT (replacing HHHH and HHHT respectively).
I would expect a less pronounced version of the same effect. Both get to HT together, but if you’re looking for HTH and get HTT then you’re starting over hoping for your first H on the next flip, whereas if you’re looking for HTT and get HTH you’ve got a small headstart because that last H can be the first H going forward to HT[HTT]