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My friend and I had an argument about whether taking a card out of order (assuming the person is not cheating, the rule is that they must take cards in order, and the deck is sufficiently random) affects the outcome of the game.

My position is that the deck has a set order/state after the shuffle (even if it is unknown) and taking a card out of order can affect the outcome regardless of knowledge of what the deck contains.

His position is that the deck order is random and so the real effect of going out of order (assuming no cheating) doesn't matter because the original outcome was unknown.

Bottom line, assuming no one is cheating, does a person have a legitimate right to question the outcome of a hand of cards if someone took a card out of order or is the order one takes cards irrelevant outside of cheating (using a stacked deck, etc.)?

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Imagine during the dealing you are offered any card, not just the top card that would normally be dealt to you. Can you profit by using this option?

No. In the long run your friend is correct. In a specific hand, yes the outcome can be different, but my opinion is you have not been wronged when a different card gets dealt to someone.

I would argue that others agree, as when a card gets accidentally exposed in a casino (my experience is in poker) the exposed card is shown to the whole table and the dealing continues. This is a situation that does affect a player's strategy. For example if a jack is exposed in hold'em and you hold JJ, you are definitely not happy. Nevertheless, the hand goes on.

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Let's say that we're playing a simple card game: one in which the person who draws, say, the queen of spades loses the game. Each player must draw one card on each of their turns.

In a game with 4 players, ($ p_1 $, $ p_2 $, $ p_3 $, $ p_4 $) seated clockwise around a table with the sequence of play continuing around the table in that order. Let's further speculate that in this current round, there have been 48 cards taken and the queen of spades remains in the deck.

We can see that, at each point $ p_4 $ would have to "cheat" the game, that is, to draw his or her card early, the probability would be the same to draw the card as if the player waited:

Before $ p_1 $ chooses (with 4 cards remaining):
$\frac34$ $*$ $\frac23$ $*$ $\frac12$ = $\frac6{24}$ = $\frac14$

Before $ p_2 $ chooses (with 3 cards remaining):
$\frac23$ $*$ $\frac12$ = $\frac2{6}$ = $\frac13$

Before $ p_3 $ chooses (with 2 cards remaining):
$\frac12$

So, we can see that in this terminal case, there would be no probabilistic difference in the outcome, assuming no cheating.

Before the cards are drawn in this specific game there is no difference between a card drawn early and a card drawn at the normal time. Be aware that even changing the number of players from 4 to 5 drastically changes the result.

In addition, this game would be very boring and deterministic, and requires no decisions to be made. In any non-trivial game, the card being drawn early could reveal information that would allow better decisions to be made etc.

A very strong argument in the game in question against allowing such moves would be that there doesn't appear to be any reason to allow it as a game move, and that the only rational motivation for such a choice would be outside the game rules (i.e. cheating).

All of that said, assuming sufficient randomization of the cards, the order in which you actually draw from the deck can change the outcome, but this change is unlikely to be predictable as long as no additional information is produced by the act of drawing.

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