# Deck card probability distribution with missing cards

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)?

It sounds like you can basically separate the cards into three piles: cards that are known to be missing from the deck (imagine they are face up), cards that are missing from the deck but are unknown (say $$m$$ of these), and the deck itself (say $$n$$ of these). You also have a list of cards that cannot be in the second pile, and hence must be in the deck. For a card that is known to be in the deck, the probability it is on top is $$1/n$$ (it is equally likely to be anywhere in the deck). For a card that you know nothing about, the probability it is on top of the deck is $$1/(n+m)$$ (it is equally likely to be in the deck in any position or in the pile of missing cards).