Probability: Shuffling and Randomization

Since Magic is supposed to be played with a randomized deck of sixty or more cards, the issue of shuffling presents itself at the beginning of the game, as well as some situations within the game that require shuffling. What constitutes fully randomizing a deck? The three most common shuffling techniques used by players are riffle shuffles, pile shuffles, and overhand shuffles. A riffle shuffle is when the deck is divided into two halves and riffled together, alternating one card from each. A pile shuffle is when the deck is dealt out into a number of piles and then grouped back up. An overhand shuffle is just taking part of the deck and moving it to another part of the deck, such as taking a few cards from the middle and moving them to the top or bottom, over and over.

Obviously the overhand shuffle is the least effective shuffle, as "clumps" of cards (that is, small groups of cards) tend to stay together more often and not get randomized in this method. The riffle shuffle and pile shuffles are indeed the methods that professional players use. Simply put, these methods randomize the deck to a much higher degree. Pile shuffles vary among players. Some like to use four piles, while some use seven. I have seen higher-end players use seven piles. This is effective because clumps are nonexistent in this method. Any group of seven cards becomes completely divided using this method. Combined with a few riffle shuffles, this randomizes a deck to a great degree.

How good is a riffle shuffle, though? To randomize a deck of 52 playing cards, it only takes 7 times(1). With 60 cards, it should therefore take approximately 8 shuffles. When many players play a game I often see them overhand shuffle once or twice, and feign a riffle shuffle. These players then draw land after land and wonder why! According to the same web site, it would take 2500 overhand shuffles in order to get the same randomization as 7 riffle shuffles! Again my analysis holds true: Riffle shuffling is a supremely effective shuffling method. When playing a game, the best thing to do is probably pile shuffle with seven piles first, in order to mix up your deck from the previous game. After that, three or four riffle shuffles are incredibly effective. Seven or eight riffle shuffles before each game can be incredibly tedious, but the existence complete randomization is still shaky. Therefore coming close to complete randomization is the best one can do.

Even though pile shuffles and riffle shuffles combine to create a good shuffling technique, there will always be the chance of drawing no land cards or all land cards. However, that is part of being randomized, and will happen every so often. Because of this, the DCI created the Mulligan rule. That lets players shuffle their initial hand into their deck at the very beginning of the game if they are very unpleased with it. The penalty is that their new hand consists of one fewer card. This is a great idea, because the chance of drawing no land twice in a row is very slim. In the next section, I will analyze such probabilities in depth using hypergeometric distribution techniques.


Footnotes:

  1. http://www.scc.ms.unimelb.edu.au/discday/dyk/shuff.html

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