Probability and Magic: The Gathering

Decision-making is a vital aspect of Magic; however, it is not the sole discerning backbone of the game. To be able to make the right decisions means to know how to determine what is the right decision. Knowing the chance that any given event will occur is what will let the player determine what his proper strategy should be. If your opponent has five cards in hand, four Islands(1) on the table, and nothing else, how do you react? Knowledge of any possible cards in his deck can help.

More importantly, knowledge of cards that are definitely in his deck is even better. If you have a vital spell you want to cast without fear of it being countered, what will your strategy be? Will it change if you know that there are four Counterspells in his discard pile? What if there are two Counterspells and three Forbids in his discard pile? Trying to determine how much countermagic he will have is a difficult task. Nonetheless, it represents a very important situation that comes up often in the game. Knowing how to play your cards is often the key to victory.

Once again, trying to determine the best strategy is where the gray areas lie. Perhaps there is no best strategy, and there are only two equally qualitative strategies. In a case such as that, only chance will determine which strategy is the right one. Your opponent is Zev, and he has two islands untapped, one card in hand, and is at 3 life. You want to win immediately, with the Lightning Bolt in your hand that will do 3 damage to Zev. If Zev has three Counterspells in his discard pile, and there are four total in his deck, should you cast the Lightning Bolt? Assuming he has no other countermagic in his deck, you want to know the possibility that the one card he is holding is a Counterspell. Before trying to determine the probability that it is a Counterspell, we first need a clear definition of what probability is.

Definition: The probability that an event will occur is equal to the number of favorable outcomes divided by the total number of possible outcomes.

First you have to count up the cards left in Zev's deck. Upon counting, you find that there are 24 cards left. Combined with the card in his hand, there are 25 cards in his deck that you have not yet seen. One of them you know is a Counterspell. The number of unfavorable outcomes in this case is 1: Zev having the Counterspell in his hand. The number of favorable outcomes is 24: The other cards in his deck. The total number of outcomes is 25. Divide 1 by 25 and you get .04, which is 4%. Zev has a 4% chance of having a Counterspell, and therefore you have a 96% chance of winning the game immediately.

What if you had seen the top five cards of Zev's deck? If you had a card that allowed this, and you saw no Counterspell in those cards, your chances would diminish of being able to a Lightning Bolt without it being countered. The number of possible outcomes would be 20. The chance of Zev having a Counterspell would be 1/20, or 5%. Now your chance of winning immediately goes down to 95%. It is not a big drop, but it all adds up. This is just something to consider when playing the game.

However, this question can obviously become more complex. What if Zev had two Counterspells left in his deck? Suddenly this definition of probability is not as useful, since we do not know the number of favorable outcomes. Hypergeometric distribution comes into play here, and is used to determine the answer to such a problem. Drawing cards out of a deck is a major probability situation in Magic; however, there are other players in the game of probability. How effective is shuffling a deck? Can a deck truly be randomized? If so, how many shuffles are necessary to completely randomize a deck?

All of these topics play a large role in the game of Magic. One more major topic, though, is the big question of chance versus skill. When playing in a tournament, you decide to use a fast red deck. You lose all three rounds and go home complaining about how you got paired against decks designed to beat you. How accurate is that statement? To find out if the decks always beat you, first you have to determine whether you played your own deck correctly. Perhaps you played your deck incorrectly and your adversaries played flawlessly. If that was the case, then you were no match for them.

Using the people I have available, I will attempt to draw conclusions on sample deck types about how much of a chance one deck has for winning against another. By then reversing who is playing each deck, new data makes itself present, since each player plays a deck differently. The player who makes the best overall decisions is the one who will win the most with a given deck. By applying that theory, I will be able to chart the statistics of decks and players using this "chance versus skill" method.


Footnotes:

  1. Island: Land. Tap for one blue mana.

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