A question over at one of the Reddit math subs asked the probability of having seen a specific card three hands in a row. It was answered and I understand the result - it's just the probability of a run of 3 where the probability of success is that of getting the specific card.
Thinking about it, I wondered how one would calculate the probability of the following:
Given a deck of $C$ distinct cards, dealing a hand of $H$ cards and noting them, then returning them to the deck, shuffling, repeat for $K$ total hands.
What is the probability of a run of at least $R$ consecutive hands out of the $K$ total hands where the intersection of the cards in the $R$-run is non-empty, that is, one or more of the same card(s) is present in all $R$ consecutive hands?