(Metagame Archive) Strength in Numbers: The Math of Mulligans

By Olav Rokne

The English philosopher and Bishop Joseph Butler once commented that probability was the guide to life. Although I remain unconvinced of this in the big picture, his attitude would have served him well when playing Vs. System.

Often, the first and most important decision of the game is whether or not to mulligan. Sadly, a vast number of players make this decision based only on a gut feeling or instinct about their Draft or Sealed decks. Is it a good hand or a bad hand? Is what you are giving up worth what you might get? Such determinations can only really be worked out through probability.

A quick search of the Internet for “probability,” “statistics,” “card games,” “CCGs,” or “TCGs” yields a shockingly large number of different ways to get entirely incorrect calculations of probability. For card games such as Vs. System, binomial distribution does not apply. Take a simple example of binomial distribution: tossing coins. The result of one toss does not affect the result of the next, which is accounted for in binominal distribution. The events are said to be independent.

However, in a TCG, selection of successive cards is not independent because the cards are not put back into the deck. Finding the odds of drawing a particular card on one draw is relatively easy. If there are 60 cards in the deck and 1 copy of the card you are looking for, any joker can tell you that you have a 1 in 60 chance of receiving that card.

It gets more complicated when the deck contains multiples of a card and you want to calculate the odds of getting 1 copy within a certain number of draws. In the long form, you should begin by determining the total number of a particular card in your deck and the number of cards you will be drawing. For this example, we’re looking for 1 of the 4 copies of “Maguffin™” in the pre-mulligan hand of a 60-card deck. (The Maguffin in question could be any card that is key to your strategy, be it Alfred Pennyworth, Total Anarchy, Xavier’s Dream, or what have you.) On the first draw, you will have a 4 in 60 chance of drawing Maguffin, so: 4 / 60 = 6.66%. You have a 93.33% chance of not having a Maguffin yet. The chance of drawing one on the second draw is 4 in 59: 4 / 59 = 6.78%. When we multiply that by 93.33%, we get 6.32%. When added to the previous 6.66%, we have 12.99%.

We repeat this process for card number 3. You have an 87% chance of not having drawn a Maguffin and 4 copies in the 58 remaining cards, so (4 / 58) x 87% = 6%. Added to our existing 12.99%, you have an 18.99% chance of having drawn a Maguffin by the third card. The fourth card leaves us with 4 copies in 57 cards and an 81% chance of not having one yet, so: (4 / 57) x 81% = 5.68%. Added to the existing 18.99%, you have a 24.67% chance.

So, you have a 24.67% chance of drawing a Maguffin in your opening 4 cards before the mulligan if you stock 4 of them in a 60-card deck.

Since you have to repeat this procedure for every card you draw, using the long form can be extremely time consuming. Thankfully, in 1657, a clever fellow named Christiaan Huygens figured out a much more elegant method of doing this. To explain his method, we’ll first need to define factorials, permutations, and combinations.

A permutation is simply the number of ways one can arrange a series of items. For example, if you have three cards, they can be put in a pile six different ways. This is because there are three options for the bottom card in the pile, two options for the second, and one option for the third—1 x 2 x 3 = 6.

A factorial is the product of every positive integer from 1 to a specific number. So, the factorial of 5 would be: 1 x 2 x 3 x 4 x 5, or 120. In 1808, Christian Camp started using the notation “N!” to mean the factorial of N, and for brevity and simplicity, that’s the symbol used here (and in most books). On a practical level, the number of permutations for a Vs. System deck of N cards is N!. For a 60-card deck, this works out to around 8.32 x 1081.

Combinations are sets within a permutation without regard to the order of those sets. In the case of a hand drawn from a deck of cards—since the order in which the cards are drawn is irrelevant and you can only draw each card once—the formula to calculate this is:

Number of possible combinations = N! / [R! * (N – R)!]

N is the number of cards in the deck, and R is the number of cards being selected from the deck.

For example, if you select 4 cards from a deck of 11, it looks like this:

11! / [4! * (11 – 4)!] = 39,916,800 / [24 * (5,040)] = 330. So, there are 330 different ways you could draw 4 cards out of an 11 card deck.

Huygens worked out what has since become known as “hyper-geometric distribution.” Essentially, it states that when calculating the chance of success, the simplest way to do so is first to calculate the number of possible combinations that would qualify as a success and then divide that by the total number of possible permutations.

This formula will give you the chance of failure, so it is important to remember to subtract the percentage from 100% to get the chance of success.

If you can do the whole process in your head, you’re smarter than I am. Thankfully, with a little help from an Excel spreadsheet (which has hyper-geometric distribution built in!), we can quickly create some tables to memorize for easy reference.

Here are the chances of drawing card X before turn Y with no mulligan in a deck of 60 cards when you have 4 copies of card X in the deck:

Turn 1: 35.14%

Turn 2: 44.48%

Turn 3: 52.77%

Turn 4: 60.09%

Turn 5: 66.53%

Turn 6: 72.16%

Turn 7: 77.04%

Since you generally have a Constructed deck built long before a tournament, you have time to work out these numbers. But in a Sealed Pack tournament, you’ll have much less information prepared, so it is even more important to have a good grasp of probability.

The decision to mulligan becomes even more crucial in Sealed Pack formats where restricted resources can be buried at the bottom of a deck by a foolish mulligan. When deciding whether or not to mulligan, one should apply the following four-part decision-making model:

1) What cards am I giving away?

2) If I mulligan, what are the chances that I will get those cards if or when I need them?

3) What cards do I want to get?

4) What is the chance that I will get those cards either in the mulligan or by the time I need them?

The first and third questions are easy to answer. The second and fourth, however, can only be answered by using statistics.

A standard 30-card Sealed Pack deck might look something like this:

9 plot twists, consisting of:

3 good pump cards

2 team-ups

1 trick/KO effect/other interesting plot twist

3 filler

21 characters, consisting of:

2 7-drops

3 6-drops

4 5-drops

4 4-drops

4 3-drops

4 2-drops

In this deck, without mulliganing, you have a 61.22% chance of hitting your 2-drop, a 73.3% chance of hitting your 3-drop, an 82.32% chance of hitting your 4-drop, an 88.83% chance of hitting your 5-drop, an 86.2% chance of hitting your 6-drop, and a 79% chance of hitting your 7-drop. Currently, as you can see, you have the best chance of hitting the 5-drop and the worst chance of hitting your 2-drop.

Knowing this, it’s safe to say that mulliganing away a hand that has a 5-drop is no big loss, while mulliganing away a 2-drop is scarier than one might suppose. One might also conclude that a fifth 2-drop might be a good idea.

Any questions? Please email me at olavrokne(at)gmail.com.


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