Odds of drawing each card from partially known subset of cards

Question

Say you have $6$ decks of $52$ suitless cards (so $24$ indistinguishable copies of each rank). From that, you're given a randomly chosen, shuffled subset $S$ of size $52$, of which half of the cards are made known to you (though how these known cards are selected is random, and you don't know how they're ordered or distributed within $S$). Call this group of cards $K$ and call the other cards in $S$ $U$. Then, each card of $S$ is revealed and discarded one at a time. My question is, after having seen $n$ cards revealed and discarded, can you compute the probability of what the $(n+1)th$ revealed card will be without enumerating all the different ways that the revealed cards could have been taken from $K$ and $U$? Here's a link to Python script I wrote that solves the problem using such an enumeration.