Deck size – Why the minimum?

I remember back when I was a kid, decks where an endless source of thinking, tuning and testing all matters of crazy ideas. Decks that milled, that played huge fatties, like [c]Force of Nature[/c], burned the opposition to the ground etc. You didn’t always feel limited to playing just a 60 card deck (back then I was scared of playing Limited formats) and sometimes my deck could swell up to way more than the minimum number of required cards. As I’ve matured and learned about such things as statistics, I’ve also come to accept that you should make the tough cuts in your deck to keep the number of cards down, if you want to increase your chances of winning. At least that’s the accepted rule amongst experienced players.

 

But why do you need to keep your deck small?

If all of our cards are equally powerful and we can always play them, it doesn’t matter how many cards you put in your deck. It may even help you to add more against certain opponents. Unfortunately this is not the case and we therefore need to ensure that we maximize the chance of getting those that gives us the best odds of winning. This is of course meant as a general rule, but not an absolute truth. I bet both you and I could come up with a niche deck build that needs more than the minimum number of cards.

Let’s assume you got a 40 card deck (I’m a Limited player :-)) and you want to get the biggest chance of drawing at least one of your five 1-drops in that opening hand. How do we calculate that? To solve this problem, we can use a statistical analysis tool known as hypergeometric distribution. When we draw 7 cards, we could end up with a hand with no 1-drop, one 1-drop, two, three, four or even five. Since we’re only interested in the chances of drawing a hand with one or more, we can calculate the chance of getting none in our hand and subtract that from 100%. That leaves us with the odds of getting the wanted opening.

The chance of not drawing a 1-drop on the first card is 35/40 = 87,5%. So a 12,5% chance of us getting it on that first draw. The chance of not drawing a 1-drop on the second card is 34/39 = 87,2%. So a 12,8% chance of us getting it on the second draw. Going through the calculations we end up with table 1 shown below. Here I’ve listed the card number drawn, the chance of not getting a 1-drop (the “failure” column) and the chance of getting one or more (“success” column).

Draw Failure Success
1 87,5% 12,5%
2 87,2% 12,8%
3 86,8% 13,2%
4 86,5% 13,5%
5 86,1% 13,9%
6 85,7% 14,3%
7 85,3% 14,7%

Table 1: The first seven draws in a 40 card deck

 

Now that we got the probabilities for each draw, we can multiply these together. This gives us a 1 – (0,875 * 0,872 * 0,868 * 0,865 * 0,861 * 0,857 * 0,853) = 0,639 or 63,9 % chance of getting one or more 1-drops in our opening hand.

If we instead put 41 cards in our deck and do the calculation again, we’ll get table 2 shown below.

Draw Failure Success
1 87,8% 12,2%
2 87,5% 12,5%
3 87,2% 12,8%
4 86,8% 13,2%
5 86,5% 13,5%
6 86,1% 13,9%
7 85,7% 14,3%

Table 2: The first seven draws in a 41 card deck

If we multiply the values we get a 62,9% chance of success. So by adding one more card, we effectively threw away a small percent of getting the desired result. We could write a similar test for any combination of deck sizes and number of cards we want to draw, you could even extend the table to look at your following draws, but the conclusion would be the same. The more cards you add the less the chance gets.

 

Conclusion

Based on the calculations done above I’ve shown that the more cards you put in your deck, the less a chance you’ll have of drawing a specific card. As a rule of thumb you should only run the minimum number of required cards, unless you have a really good reason to do otherwise.

 

Further reading

If you’re interested in reading more from some of the best Magic players on the theme of deck size, I highly recommend the two articles below. These have served as inspiration for me.

Patrick Chapin: 61 Cards – Magic Russian Roulette

Frank Karsten: Is Playing More Than 60 Cards Always A Bad Idea

2 thoughts on “Deck size – Why the minimum?

  1. I totally understand the decision to limit you to 4 copies of any one card; this needs no explanation. However, why did they change the minimum deck size? I can find conflicted references to

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