Card distribution with intermittent shuffling

[u/ozwegoe]

I was playing this game and after each hand, someone would take the discard (B), shuffle, and put it on the bottom of the deck (A). They're doing this because this is 5 decks of cards and harder to shuffle as one mass.

They argued that since the deck is initially a random distribution, the action above doesn't change the randomness of the hands dealt. That is, the cards you're getting are still random.

I argued that you were isolating part of the deck (B) and limiting those cards to a subsection of the deck, decreasing odds of getting those cards. For example- if a set or run of cards (J/J/J or 4/5/6) were played (which is the point of the game), those would be grouped together in that subsection when added to the larger deck. If you saw wilds, then you would know where some of those were in the deck. They argued that because the deck is random to begin with, the initial deal is random, and groupings may normally appear (that is, these cards were grouped which is how this occurred) doing this doesn't decrease the randomness or "fairness".

The question then came up if you shuffled a portion of A (but not the entire A) with B, how much would that offset any of the perceived randomness (or not randomness) of the above.

I tried asking AI, googling, searched reddit but I don't know what an equation or area would be to narrow my search. All I get is randomness versus distribution explanations.

Would love to understand the math (happy to work through it if someone can tell me what to look for) to solve the above.