Probability: Drawing Cards From a Deck

Now the question arises again: If you have four of a card in your sixty-card deck, what is the chance you will draw one in your opening hand? If you had only one copy of the card, the problem becomes simple. Probability is defined as the number of chances that the event will occur divided by the total number of possible outcomes. Therefore if you had one copy of a card in your sixty-card deck, you would have a 7/60 chance of drawing it in your initial seven-card hand. If you have four copies of the card, though, does the chance become 4/60? Or perhaps it is more or less than that quantity?

First let me do a quick review of some math concepts. Factorials, permutations, combinations, and hypergeometric distributions are much of what make up this topic. Before I start referring to these concepts I will first explain each of them so the non-mathematically motivated reader can catch up.

A factorial is calculated by multiplying all the whole numbers from 1 up to that number together. It is written in the form "N!" where N is a positive number. For example, 1 x 2 x 3 x 4 x 5 = 120 = 5! which is read as "5 factorial". Zero factorial is defined as 1. While it does not immediately make sense mathematically speaking, trying to find the number of permutations available for zero items makes sense; there is only one way to arrange zero items.

A permutation represents the number of different ways one can arrange N objects. If you have 5 books and want to see how many different ways you can stack them, you would use a factorial. The first book would have 5 possible positions; the next one would have 4 possible positions left, and so on. Therefore there are 120 (5!) ways to arrange 5 books. If you only wanted to arrange 3 books, you would only need to compute 5 x 4 x 3 which is equal to 60. That is because you are only using 3 positions.

A combination simply represents a number of possible subsets, and is not concerned with the arrangement of these items. If you were to choose 3 out of 5 books, how many different combinations are possible? For this, you would take the number of permutations, but then divide by the number of repeated combinations. You do this by taking the factorial of the number of items. For this example, you would find 5 * 4 * 3 and divide it by 1 * 2 * 3. Therefore 60 divided by 6 is 10, and there are 10 different ways to pick 3 out of 5 books.

A hypergeometric distribution is something much more complex. It is used to determine the probability of certain sets of occurrences when extracting elements without replacement. That definition certainly applies to drawing cards from a deck, since you are taking them out. Hypergeometric distribution may seem like an unfamiliar phrase, but it is a concept that we are all fairly familiar with. When drawing cards from a deck without putting them back, this concept applies. This formula can be used to determine how often you draw certain cards from a deck of cards.

The formula syntax is a bit complex, though. Recall the formula for the number of combinations for 3 out of 5 items, that is C (5, 3) is (5 * 4 * 3 / 1 * 2 * 3) or 10. Alternatively, this can be written as 5! / (3! * (5-3)!). This can be converted into the general notation X! / (Y! * (X-Y)!). X is the total number of items to choose from, and Y is the number being chosen. Using the same X and Y notation, the formula for hypergeometric distribution H (X, Y) is as follows:

H (X1 Xn, Y1 Yn) =

C (X1, Y1) * * C (Xn, Yn) / C (X1 + + Xn, Y1 + + Yn)

However, this can be greatly simplified instead of having to go through each item from 1 to N, where N is the total number of cards. In a two-set case, that is, all the cards you are concerned about being one case and the rest being the other, this is the simplified formula:

H (n) = C (X, n) * C (Y - X, Z - n) / C (Y, Z)

X stands for the number of a certain card that you have in the deck.

Y is the number of cards in the deck.

Z is the number of cards you are drawing.

N is the number you are checking for.

Instead of doing all this arithmetic by hand or with a super calculator that can handle such large factorials, a spreadsheet such as Excel can be used to find hypergeometric distributions. The syntax is HYPGEOMDIST (N, Z, X, Y). For instance, if you have a 60-card deck, what will be your chances of not drawing one of your 4 Lightning Bolts on turn 1? By using HYPGEOMDIST (0, 7, 4, 60) you will get the chance for not drawing the card. Therefore if you want to check for the chances of drawing a Lightning Bolt, you would subtract the result from 1.

Turn 1 60.05%
Turn 2 55.52%
Turn 3 51.25%
Turn 4 47.23%
Turn 5 43.45%
Turn 6 39.90%
Turn 7 36.58%
Turn 8 33.46%
Turn 9 30.55%
Turn 10 27.84%

Those are the chances of not drawing a Lightning Bolt. On turn 1, you will have 7 cards, and there is a 60.05% chance that you have not drawn one of your 4 Lightning Bolts out of 60 cards. By turn 10 this chance diminishes to 27.84%. Likewise, the chance of drawing one or more by turn 1 is 39.95%, and the chance increases to 72.16% by turn 10.

Hypergeometric distribution has other useful applications within the game. Calculating how many land cards to use in a deck is the base of deckbuilding, as one needs land in order to play his cards. Too many land cards will cause you to draw not enough good cards late in the game, and too few will cause you to stall, giving your opponent the advantage. Some decks will want more land and be able to take advantage of it, and others will want fewer because of smaller mana requirements. If you want to draw four land by turn four often, but not too often, this formula is helpful.

The best thing to do within the game would be to have a 70-80% chance of this happening, then use cards that allow you to look through your deck to get more. Right now I will examine the chances of drawing four land cards by turn four if there are varying amounts in the deck. Using HYPGEOMDIST on Excel, this takes a few steps. For each number of land in the deck (I'll use 16 through 30), you have to determine the chance of drawing 0, 1, 2 and 3 land in 10 cards out of a 60-card deck. Then they have to be added up, and the total subtracted from 1. The final numbers are the chances of drawing 4 land by turn 4.

16 24.99%
17 29.52%
18 34.25%
19 39.12%
20 44.05%
21 48.98%
22 53.85%
23 58.60%
24 63.18%
25 67.54%
26 71.64%
27 75.46%
28 78.97%
29 82.17%
30 85.05%

Depending on what ratio you are willing to work with, it would appear that 28 land seems to give a fair chance of drawing the lands this deck needs. Depending the number of cards in the deck that allow the player to look through extra cards, anywhere from 24 to 28 land is the number to use. Cards such as Impulse(1) help remedy this cause. As long as the player has 2 land to cast it with, Impulse can get the player what he needs anytime in the game.


Footnotes:

  1. Impulse: Instant, 1U. Look at the top 4 cards of your deck. Put one in your hand and the other 3 on the bottom of your deck in any order.

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