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This question is not related to "Ticket to Ride", but while playing Ticket to Ride, the following happened:

There were 5 us around the table, and to make it easier to reach for all players, we split up the draw deck into 2 decks, one on each side of the table.

Is that okay? In theory, drawing a card from the deck is meant to be "random", so drawing from anywhere in the deck is still "random".

On the other hand, once the cards have been physically shuffled, they are what they are, then it becomes more of a "fate" in regards to drawing.

So I wanted to get a response from others: Is it okay to draw from the middle of the deck?

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    This question is essentially an opinion poll. Some people will say it's ok, some will say it's not. Mathematically it does not matter which card your draw; anything to be said further on this matter is metaphysical. – singletee Jun 2 '16 at 14:29
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    Note that for Ticket to Ride specifically; destination cards that are returned go to the bottom of the pile, and could come back up again, so you can't do this with the destination cards. But for the train cards; there's no difference between drawing the top card or the middle card. – GendoIkari Jun 2 '16 at 15:15
  • I'd say drawing from the middle of the deck is prohibited, but splitting the deck in two and then drawing from the top of either deck is allowed. – Alexander Jun 2 '16 at 18:13
  • @Alexander But what is the difference? Splitting the deck and drawing from either is essentially the same thing? – ardavis Jun 2 '16 at 18:32
  • @singletee - that would be true if the deck were perfectly shuffled. Be honest... On the 3rd go through the train cards, do you really shuffle well enough that they are perfectly ordered? – Scott Jun 2 '16 at 23:42
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Mathematically the probabilities are unaffected. Let's consider a simplified example with just 4 cards.

Let's say we draw one card from the 4 aces from a standard deck of playing cards. The probability of drawing the Ace of Spades is 1/4.

Now let's split the deck evenly and pick the first card from the first split to compare the probability of drawing the Ace of Spades. Using the Law of Total Probability, there is half a chance the Ace of Spades is in the first split of 2 cards. There is also half a chance that in the first split, the first card is the Ace of Spades.

Multiplying these out gives the probability of drawing the Ace of Spades as being (1/2)*(1/2) = 1/4, just as in the case of drawing normally.

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    Unfortunately, for most people, "mathematically true" and "acceptable answer" are different (see also Monty Hall Problem). People are predisposed to the cards being "ordered", even if the order is indeed random. – hymie Jun 2 '16 at 14:52
  • Yes, I once had a very serious argument with my wife's family after I defended someone who was drawing Carcassonne tiles ahead of his turn (so that he could play faster when it came around). There's literally no advantage to him as long as he doesn't table talk, but my mother in law and sister in law did not see it that way. – Andrew Vandever Jun 2 '16 at 17:17
  • @AndrewVandever Just had the same thing with Ticket to Ride recently, where I suggested that a player can go ahead and take their turn while the previous player is spending time figuring out which newly-drawn destination tickets to keep (the way I've always played). While there's a very slight knowledge advantage to the player deciding on the tickets; if the player who would be giving that knowledge is ok with it, it should be fine. – GendoIkari Jun 2 '16 at 17:30
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    @GendoIkari While the information gained by the person deciding on the tickets usually won't matter, there's definitely the possibility that it could, if the next person plays on a connection critical to one of the tickets in hand. It's a low probability of significant knowledge, which on average is a very slight knowledge advantage, but it seems qualitatively different. I'd let the next player draw cards, but if they're going to play trains, I'd wait until the tickets are selected. – Nuclear Wang Jun 2 '16 at 18:50
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    The other element to consider is that some people will consider playing ahead impatient and rude, regardless of any practical implications. Setting expectations as a group is key here. – Andrew Vandever Jun 2 '16 at 19:06
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From the Ticket to Ride official rules:

Draw Train Car Cards – The player may draw 2 Train Car cards. He may take any one of the face-up cards or he may draw the top card from the deck (this is a blind draw). If he draws a face up card, he immediately turns a replacement card face-up from the deck. He then draws his second card, either from the face up cards or from the top of the deck. (See Train Car Cards for special rules for Locomotive cards).

So for this specific game, no, it's not allowed to draw from the middle of the deck.

The chance of drawing a specific card is the same no matter what card you draw so it dosen't matter. But if it dosen't matter, why go through the trouble of drawing the middle card?

As long as the game dosen't interact with the deck (like in MTG where you can peak at the cards, or order them) there's nothing wrong with creating your own house rules and draw whatever card you'd like.

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    "why go through the trouble of drawing the middle card?" - because, as the question states, they wanted to make it easier for players to reach the deck, and thus split it in two, making the top of one of those stacks actually the middle of the deck. – Andrew Vandever Jun 2 '16 at 17:19
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    I would rephrase "As long as the game dosen't interact with the deck" as "As long as players do not gain knowledge of cards in the deck." It's more direct with the exact issue that invalidates the Law of Total Probability as stated in the other post. – Andrew Jun 2 '16 at 19:21
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    The rules of the game also state that "The player who is the most experienced traveler goes first." Other games have similar rules, yet in almost all games I've played, the starting player is picked by random draw. – ilkkachu Jun 2 '16 at 21:10
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As long as you draw the whole deck before shuffling discards, it doesn't matter. Simply put, all cards in a well shuffled deck are equal. (Or have an equal probability of being this or that.)

Splitting the deck, drawing in the wrong order, or intentionally drawing from the middle of the deck, or even shuffling the deck between draws does not matter, if there's no way to gain knowledge of the cards, either from how the deck is used in the game, or by (accidental) peeking.

However, in games where (e.g.) used or discarded cards are put back to the bottom of the deck without a reshuffle, it obviously matters. (The mechanic is usually something like draw two, keep one, put the other to the bottom of the deck.)

Shuffling the discards in to the deck before all cards are drawn once, will also change the odds slightly. Also, in some games, players may actually want to draw cards from the original deck (before drawing cards that have been discarded and reshuffled once), so when the end of the deck is near, even minor rules about the order of almost simultaneous draws will matter.

Incidentally, since Ticket to Ride was mentioned: well shuffled is also a key word, since a badly shuffled deck might very well have bunched remains of previous sets.

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Of course, any of that doesn't stop people from being superstitious. :) I'll admit to sometimes drawing from the middle of a shuffled deck in cases where my previous draws (from the top) have been utter garbage. I know it doesn't change the odds (and if it did, it would be cheating), but only makes me feel better by being able to act out on my misfortune. ;)

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It's true that there's no statistical difference between a card taken from the top of a perfectly shuffled deck and a card taken from the middle of the deck. (Assuming that no one has already looked at the top cards.)

The caveat is perfectly shuffled. Humans, in general, are rather bad at shuffling cards. For a standard 52 card deck, it takes about 7 riffle shuffles to randomize the deck well. Very few people will shuffle that many times. This means that there is normally some residual order in the deck, even after shuffling.

People have been know to take advantage of that. For example in bridge, top players can (unconsciously) incorporate the non-randomness of typical shuffling patterns into their playing decisions. I remember hearing that when automatic shufflers were first introduced for bridge tournaments, there were objections that the machines weren't shuffling "properly", as the randomness of the decks were greater than the previous human shuffles, throwing of established strategies.

So that is something to consider. If you don't have a truly random shuffle in the deck, there may be some residual order in the cards. If you then split the deck into multiple draw piles, this would allow a particularly perceptive individual to gain an advantage by strategically drawing from whichever deck is "hot" at the moment. You can, of course, mitigate this by preventing people from picking which pile they draw from (e.g. Forcing everyone to draw from the closest deck), or simply shuffling better.

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First, there are games where rearranging/peeking at the top cards is an inherent mechanic, so obviously, don't split and draw in those games.

Furthermore, splitting the deck before drawing might be perceived as a hint that card backs are marked, and that you actually know which card is coming, and you don't want it.

For those reasons, and the superstition of some players, I'd go against splitting decks unless everyone at the table agrees beforehand, and the game mechanics allow it.

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