Unknown Card Monty Hall

Question

I'm trying to understand this problem better and whether or not it is defined as a variant of the classic Monty Hall problem.

Let's say there are X cards in a deck of cards. A copy of this deck of cards is used by everyone, so only 1 player uses this deck. There are Y cards labeled "A". Sometimes the cards are dealt face up, sometimes face down. When they are dealt face down, the receiver of the cards is unaware whether the card is "A" or some other card. The cards are used to track information and the cards labeled "A" are just one of several types of cards. The player does not have the option to draw them face up or down and cannot change how they are drawn at any point.

Now, there are certain circumstances where additional cards are to be dealt beyond what is required. The argument comes from whether drawing the extra card(s) changes the distribution of the remaining cards and the probability that a card labeled "A" changes if the cards are drawn. Again, the player does not always draw the cards face up and does not always draw additional cards. The player has no knowledge of the cards unless they are drawn face up.

So, does drawing or not drawing the extra cards have an effect on the probability of the remaining cards? Is this a Monty Hall problem? I didn't think it was because the only a priori information available to the player is cards that have been drawn face up and how many cards have been drawn (whereas Monty knows which door the car is behind and the player can see the result of their initial selection and is given the option to change their selection).

Answer 1

Monty hall is a situation where you KNOW the distribution of A and B among the original doors, i.e. one is A and the other two are B. Also, it is always given that a door labeled B will be revealed (without being chosen) before the player gets to decide whether to switch or not. Your problem does not have these properties. If you knew how many A were in the deck, then yes you could say the probability changes based on what cards have already been dealt. But if you have no knowledge about the original distribution of cards, then for all you know all the undealt cards are all A or they are all B.

Answer 2

It is not a Monty Hall problem.

The cards follow a Hypergeometric distribution. The best predictor of a Hypergeometric distribution is that it follows the distribution of the cards that have been revealed. For example, if $N=100$, and 10 cards have been dealt face down and 10 face up and the face up cards are 3 "A" and 7 not "A" then your best assumption is that 3 of the face down cards are "A" and 24 of the undealt cards are "A". This means that if you draw other cards your best expectation is that you will receive the same proportion as you already have.

Revised following OP comment

Since you know the original distribution, your knowledge does change your understanding of the dealt cards.

If there are $n$ cards with $a$ of type "A", and $n_1$ of these are dealt face up of which $a_1$ are type "A", then the remaining cards are hypergeometric with $N=n-n_1$ and $k=a-a_1$.

Answer 3

I don't have the reputation necessary to comment so I'll put this here instead.

The question OP actually meant to ask is the following...

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You have a deck of $N$ cards, $Y$ of which are labelled "A". $N$ and $Y$ are known.

Does discarding $n$ cards face down from the top of the deck affect the probability that the next card is labelled "A"?