Brewer's Minute: Hypergeometric Distribution in Deck Building

Hey, everyone! It's time for another Brewer's Minute. Deck building in Magic, beneath everything else, is a numbers game. How many lands do you play? How many sources of each color of mana do you need to play your spells on time? How often will you find your finisher on the right turn? Most Magic players depend on intuition and experience when it comes to making this all-important decision, and while this style of deck building is fine, there is another more scientific way: math. With the help of the hypergeometric distribution, we can answer all of these questions in a in a concrete way, without relying on our intuition or making educated guesses. While this probably sounds intimidating, and more so if you aren't a math person, it's actually pretty easy, especially with the help of a hypergeometric calculator, which does all of the hard work for us!

We'll get to the video in just a minute, but first a heads up. Normally, I do an article that's essentially a transcript of the video for those of you who don't have the time to watch, but this week, we can't really do a full article because you really need to see the examples in the video to understand what's happening, and it would be impossible to write out all of the math in a way that makes sense. Instead, we'll have a breakdown of some resources and a general breakdown of how to use the hypergeometric calculator.

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Discussion

Above is a picture of the hypergeometric calculator that I use (if you want to play around with it, you can find it at StatTrek.com). While I'm sure there are other options available as well, this is the one I'm most familiar with, so it's the one we'll use today, but if you have a favorite, feel free to use it—the answers should be the same.

Now, I realize that this probably looks intimidating, but in reality it's simple. To run a calculation, we just need to put in four numbers, and the calculator will spit out an answer. Let's start by talking about what each value means in terms of Magic, and then we'll run through an example so you can see how the calculator works in practice (for more examples and uses of the hypergeometric calculator in deck building, make sure to watch the video). 

Population Size: This one is simple—here, we enter the number of cards in our deck. So, if we are calculating the odds of drawing something in our opening hand, it would be 60 for most constructed decks, 40 for limited, and 99 for Commander. Of course, this changes as the game goes along. If we are on Turn 10 and have only 42 cards left in our library, we'd enter 42 rather than 60. 

Numbers of Successes in Population: This is basically the cards or group of cards we are interested in. If we are trying to calculate the odds of having one of our four Aetherworks Marvels in our opening hand, we'd enter 4. If we are calculating the odds of drawing two of our 20 Islands, we'd enter 20. If we are calculating the chances that we hit one of our three Ulamog, the Ceaseless Hunger with our Aetherworks Marvel, we'd enter 3, and so forth. 

Sample Size: This is the number of cards we are drawing. If we are calculating our opening hand, it would be 7. If we are calculating the odds of a top deck, it would be 1. If we are calculating the odds of hitting something with our Aetherworks Marvel, it would be 6 (the number of cards we look at with Marvel). 

Number of Successes in Population: This relates back to the card or group of cards we talked about before. If we care about having one Aetherworks Marvel in our opening hand, we enter 1. If we want to know the odds of having two of our 20 Islands in our opening hand, we'd enter 2. If we wanted to know how often we'll not have an Ulamog, the Ceaseless Hunger in our opening hand, we could even enter zero!

So, let's run through a quick example, which will give us a chance to talk about the answers we get from the calculator. Let's use something simple. We are playing Temur Marvel in Standard (so a typical 60-card deck). We have four copies of Aetherworks Marvel, and we want to know the odds of having a copy in our opening hand. If we plug this into the calculator, it looks like this. 

Finally, let's take a look at what the calculator looks like with the answers filled in, and then we'll talk about what these answers mean. 

First, the hypergeometric probability is the odds of having exactly one copy of Aetherworks Marvel in our opening hand (which is 33.62%, so basically one in every three games). However, some of the other numbers are more useful in regards to Magic because they account for a wider range of outcomes. For example, the very bottom number is especially important to the Aetherworks Marvel example because it not only include the 33.62% of hands that include exactly one Marvel but also the (roughly) 6% of hands that have two or more Aetherworks Marvels, which raises the percentage up to just under 40%. 

Of course, this is a very simplistic example, and we talk about a lot more in the video, but let's wrap up the article with one more quick calculation. When it comes to Aetherworks Marvel, we don't really care about having a copy in our opening hand because we can't cast it until Turn 4 (most commonly) anyway. So what if, instead of looking at our opening hand, we look at our odds of having an Aetherworks Marvel by our fourth turn of the game? For this, we make one little tweak of the numbers. While we can leave everything else the same, let's change the sample size to 11 (which would simulate not just our opening hand of seven cards but four turns of drawing one card per turn). One note here: be aware of the differences in play / draw. Having 11 draws by Turn 4 is correct for being on the draw, but this number would drop to 10 on the play. 

As you can see, if we look at Turn 4 instead of our opening seven, the odds of having at least one Aetherworks Marvel jumps all the way to 56.55%. So, next time you're opponent slams an Aetherworks Marvel on Turn 4, they didn't get lucky; instead, they are actually (slightly) unlucky when they don't have a Marvel by Turn 4!

Conclusion

Anyway, that's all for today. Learning how to use a hypergeometric calculator is invaluable to building functional decks, and the possibilities are endless! Make sure to watch the video for some more examples, and if you have any questions, make sure to let me know in the comments. As always, you can reach me on Twitter @SaffronOlive or at SaffronOlive@MTGGoldfish.com.