Once again, I’m probably not gonna read everything in this topic, including your replies to stgange posts — thus I’m sorry if I missed something meaningful (by the way, the maths in your post doesn’t look very readable to me, sorry; even if you used LaTeX notation everywhere, that’d probably be easier for me to understand). Nevertheless, lemme point out something related to the substance of the matter:
I don’t get it.
If you wanna account for pity timers (good point, I haven’t thought much of it), you probably gotta use conditional probabilities. For example:
P(X legendaries) = P(hitting 0 pity timers) * P (X leg. | 0 pity timers) + P(hitting 1 pity timer) * P (X leg. | 1 pity timer) + P(2 pity timers) * P (X leg. | 2 pity timers) + …
Admittedly, it might get cumbersome, but it looks like a proper rigorous way to account for pity timers to me. Am I missing something?
Apart from this little problem, there’s another non-mathematical aspect where I think you’re making another mistake (the first one being that ‘stepping of the same rake’ with those packs, as discussed above) — I hope you don’t mind my pointing it out. Namely, you’ve chosen to embark on conversations with very strange characters (persons or chatbots, I dunno) seemingly producing nothing but meaningless ‘word salad’ (because banana phlegm, farthing flymum…) of supposedly inflammatory (?) nature — and in large quantities at that. I don’t think this is going to take you anywhere if you wanna discuss your results or calculations (those forum ‘pundits’ also tend to have a zero mathematical aptitude, as well as an intellectual one in general — why, not even something in the department of common sense and decency, it would seem… but I digress a bit). You’re your own man, of course, but please don’t tell me I didn’t warn you 
if this topic results in ten more pages of:
Maragibo, safluses!
Hergling flur dianomomo.
Coconut… boot… kahoogling.
Ah, cowda hericken zar zar.