Probability: The Average Game

In a game of Magic, there is a lot of decision making and probability. I have already given a glimpse of the decision-making aspect. Now I shall examine the probability aspect. Several random events happen in Magic. The cards you draw are somewhat a factor of randomness. There may be games that you get the same cards, but rarely in the same order. Often times the game might play almost identical to another game because of deck searching cards used to get certain cards, but two games are rarely identical to one another. Deck randomization is one major factor in probability affecting the game. If you need to draw a game-altering card in order to seize advantage of the game, you have to determine the chance of drawing that card and plan ahead for another strategy should you not draw the card you need. Having a backup plan is fail-safe, simply because relying on a single strategy that will not always work will not get you very far.

There are a few cards in Magic that make you flip a coin, but it is rarely an issue within the game. A couple of years ago there was a coin-flip card that saw plenty of tournament play called Frenetic Efreet(1). It utilized a game mechanic called phasing. When a card phases out it is removed from the game. It phases back in during its controller's next untap phase. This card was good because if it was the target of a card such as Lightning Bolt or Shock, its controller could attempt to save it by phasing it out. It would have a 50 percent chance of being lost as opposed to a 100 percent chance of being lost if it lacked its special ability. Over time, this ability would net its controller an extreme advantage. If the opponent had to use two cards in order to remove a single card, that meant that the player who only lost one card had more resources left. Such card advantage is crucial to gaining an advantage in the game.

Cards like Wrath of God, which can make the opponent lose more than one card, are powerful. When playing more than one creature against a deck with Wrath of God in it, you need to try and figure out whether it is better to play more creatures and try to defeat the person fast, or to anticipate a Wrath and play creatures after. Figuring the probability of a Wrath is useful. Just knowing some basic figures, such as the chance of the opponent drawing one after 5, 6, 7 turns is all a player should be comfortable. Doing hypergeometric distributions on the spot is definitely not something a player needs. Common sense usually dictates the right play.

If there is one card in a deck that can give the deck a severe advantage against an opposing deck, having drawn that card at the right point in the course of a game also adds to randomness. If the card is needed early, getting it within the first three or four turns will often be a deciding factor. This chance is around fifty percent, given there are four of the card in a deck, as shown through hypergeometric distribution.

The largest random factor in Magic is by far randomness of the deck. The cards a player draws from his deck are chosen at random. Therefore what card a player is going to draw is an unknown until he draws his next card. Not having this knowledge limits a player's ability to think and plan ahead. Compare this to the popular game of Tetris. In Tetris, the object is to stack falling blocks so they arrange into lines without gaps. By viewing the next piece, the player easily plans a strategy, and plans ahead in order to succeed. In Magic, the player does not know his next card, and can only plan in the present. That is, unless the player has a card in play which allows him to view the next card or cards in his deck. Generally, being able to plan ahead is limited to the cards in one's hand. Knowing the chances of what cards will be drawn later in is only sketchy, because the player cannot determine exactly when the cards will be drawn.

Take the following situation. You are in a duel with Andrew and you are at 8 life. Andrew is playing a fast black deck with lots of small creatures that are slowly diminishing your life total. You are playing a blue and white control deck and have altered your deck between games (this is called sideboarding(2)) in order to help you defeat Andrew's deck. He has not put in any cards from his sideboard. In order to guarantee your victory against Andrew, you have put 3 copies of the card Light of Day(3) in your deck.

Since you know the entire contents of Andrew's deck, you know that Light of Day will stop him cold in his tracks and let you do whatever you want, since his one method of victory will be gone. It is currently the end of your seventh turn. You played first this game. You still have not drawn a Light of Day, but have all the land you need for it: 4 Islands and 3 Plains. However, if you draw one of your 4 copies of Intuition(4) you will be able to search your deck for a Light of Day and play it immediately. What are the chances of you winning next turn? What are your chances of winning the turn after? What do you do in the meantime if you do not draw a Light of Day?

Since you have taken 7 turns and played first, and we will assume you have used no deck manipulation cards, you have gone through 13 cards. Of those 13 cards, none were Intuitions or Light of Days. Since there are 4 Intuitions and 3 Light of Days, you have 7 favorable outcomes out of 47 cards that can be drawn from next turn. For the turn after, you will have 7 out of 46 if you do not draw a key card next turn. That does not mean a chance of 7/46, though. Using the formula from the last chapter, you would find HYPGEOMDIST (1, 1, 7, 47) because you are checking for 1 card, you are drawing 1 card, you have 7 favorable outcomes, and 47 total outcomes possible. Your chance of winning next turn is 14.9%. To find the chance of drawing the card the turn after if it is not drawn next turn, change the 47 with 46 since there would be one other card drawn next turn which is not directly useful.

To find the total chance between the two next turns, you would need to find the sum of HYPGEOMDIST (1, 2, 7, 47) and HYPGEOMDIST (2, 2, 7, 47) since you are drawing 2 cards, and there can be either 1 or 2 of your 7 key cards drawn. Your chance of winning in two turns is 27.8% if you do not win next turn. What is your chance of winning in 3 turns from now if you do not win within two turns?

By adding up the chances of drawing 1, 2, or 3 copies of your key cards by then, the answer comes to 39.0%. Of course, that is the chance in advance. If after two turns you have not drawn any of the key cards, your chance is actually lower. The same applies to the previous example for winning in two turns, after drawing the first card. Then, the chance is similar to the first example, with one card being drawn. If you do not win after one turn, your chance of winning on the second turn is 15.2%. If you do not win on the second turn, your chance of winning on the third turn increases to 15.5%. This is found by taking the hypergeometric distribution of 46 and 45 cards, respectively, instead of 47.

If you find that these chances are too low, you will need to plan an alternate strategy. Such a strategy might include using your available resources to stalling out Andrew instead of saving them to win the game immediately after you get your lock. For example, you might play a creature now and use it to block one of his in the meantime in order to save you some damage. There is no clear method of probability making the right decision. Instead, a player needs to use probability to plan a strategy. A strategy will provide decisions for any given situation.


Footnotes:

  1. Frenetic Efreet: Creature, 1RU. 0: Flip a coin. If you win, it phases out. Otherwise it is buried. Flying. 2/1.
  2. A word about sideboards: In a tournament, players play the best two out of three games. Between games 1 and 2, and between games 2 and 3, each player can exchange on a one-for-one basis cards between his deck and 15-card sideboard. Players are allowed to use a sideboard optionally; either a 0 card or 15-card sideboard is used.
  3. Light of Day: Enchantment, 3W. Black creatures cannot attack or block.
  4. Intuition: Instant, 2U. Search your deck for any three cards and show them to your opponent, who then chooses one card. The chosen card goes in your hand, and the other two in your discard pile.

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