Given a deck of cards (standard 52 deck), we know that there are some cards missing. Some of the cards that are missing we know exactly what they are (e.g. an Ace, a Two are missing). Other cards that are missing we just know they are not some card (e.g. a card is missing and it is not a Three, or a card is missing and it is not a Ten).
For convenience we can represent the cards missing from the deck as two arrays or vectors:
Known missing cards (Ace-0, Two-0, Three-0, ..., King-0) - (These are the cards that are missing that we know exactly what they are.)
NOT missing cards (Ace-0, Two-0, Three-0, ..., King-0) - (These are the cards missing that we don't know what they are exactly, but we know they are not some card.)
For example if we have
Known missing cards: (Ace-0, Two-2, Three-0, Four-1, Five-0, Six-0, Seven-0, Eight-0, Nine-0, Ten-0, Jack-0, Queen-0, King-0)
NOT missing cards: (Ace-3, Two-0, Three-0, Four-0, Five-1, Six-0, Seven-0, Eight-0, Nine-0, Ten-0, Jack-0, Queen-1, King-0)
We know that the deck has 2 Twos missing, 1 Four missing, 3 cards missing that are not Aces, 1 card missing that is not a Five, and 1 card missing that is not a Queen (8 cards missing total).
Given this information, in the general case (with arbitrary arrays), what is the probability distribution of the next card in the deck?
Also, a follow up question to this is what is the probability distribution for each of the missing cards that we know are not something? For example, if we know a card is missing and it is not a Two, what is the probability distribution of that cards value (given the arbitrary arrays)?