If LLMs are anything like people, I would expect a different result depending on that. The idea that random events are independent is very unintuitive to us, resulting in what we call the Gambler's Fallacy. LLMs attempts at randomness are very likely to be just as biased, if not more.
I think randomness needs to be better defined. In the article it seems to be that randomness should be an evenly distributed type of event occurences.
I agree that it is very unintuitive for us as, I believe, we assume randomness to be any sequence of event that doesn't follow any known/recognizable pattern. Show a section of the Fibonacci to a 10 yo kid and they will most likely find the sequence of numbers to be random (maybe they will note that it is always increasing, but that's it).
Even in this article the fact that o1 always throws "heads" could indicate that it "knows" what randomness is, and is then just being random by throwing only heads.
I personnaly would define ideal randomness as a behavior that is fundamentally uncomputable and/or cannot be expressed as a mathematical function. If this definition holds than the question cannot apply to LLMs as they are a just (big) mathematical function.
If LLMs are anything like people, I would expect a different result depending on that. The idea that random events are independent is very unintuitive to us, resulting in what we call the Gambler's Fallacy. LLMs attempts at randomness are very likely to be just as biased, if not more.