I had to write several new Python versions of the code to explore the problem before it clicked for me.
I understand the proof, but the closest I can get to a true intuition that B is bigger is:
Imagine you just rolled your first 6, haven’t rolled any odds yet, and then you roll a 2 or a 4.
In the consecutive-6 condition, it’s quite unlikely you’ll end up keeping this sequence, because you now still have to get two 6s before rolling any odds.
In the two-6 condition, you are much more likely to end up keeping this sequence, which is guaranteed to include at least one 2 or 4, and likely to include more than one before you roll that 6.
I think the main think I want to remember is that “given” or “conditional on X” means that you use the unconditional probability distribution and throw out results not conforming to X, not that you substitute a different generating function that always generates events conforming to X.
I had to write several new Python versions of the code to explore the problem before it clicked for me.
I understand the proof, but the closest I can get to a true intuition that B is bigger is:
Imagine you just rolled your first 6, haven’t rolled any odds yet, and then you roll a 2 or a 4.
In the consecutive-6 condition, it’s quite unlikely you’ll end up keeping this sequence, because you now still have to get two 6s before rolling any odds.
In the two-6 condition, you are much more likely to end up keeping this sequence, which is guaranteed to include at least one 2 or 4, and likely to include more than one before you roll that 6.
I think the main think I want to remember is that “given” or “conditional on X” means that you use the unconditional probability distribution and throw out results not conforming to X, not that you substitute a different generating function that always generates events conforming to X.