Introduction
Hello everyone, and welcome to the second installment of Marvelous Math! In these articles, we take a look at ways to use data, probability, and statistics to win more games! The goal of these articles is to help you learn more about the core aspects of the game, which will help you to gain a competitive edge over your opponents!
In today's article, we are going to take a deeper dive into draw probability and how it can influence deck-building decisions. If you haven't already, feel free to check out my first article, which gives a basic overview of probability in Marvel Snap.
Let's jump right in!
Draw Probability
Drawing cards is an important part of any card game. In order to win a game, you generally want to draw into your important cards. These may be strong early game cards to fight for the board or pieces of a strong combo you can use later on. Regardless of your game plan, drawing into the cards you want drastically increases your chances of winning.
However, drawing cards is inherently random. I am sure many of you reading have been in a situation where you needed a card in order to win and just didn't draw it, or maybe you had one last draw left and pulled the perfect card off the top of your deck! This is one of the main challenges with deckbuilding. In any deck, it is important to try and optimize it in order to get the cards you need to win more often.
This problem poses many questions. How many cards of a particular cost should I run? How often am I drawing this two-card combo? Do I need to build around a certain curve?
Let's start with an easier question: How often will I draw a specific card in a given game, ignoring outside effects? This is a pretty simple probability problem, as we just need to divide the number of cards drawn by the number of cards in our deck.
Suppose we want to have Squirrel Girl (118) in our opening hand. How often is that happening? Assuming we aren't running other cards that affect our draws, we will start with 3 random cards in hand and draw 1 card at the start of the turn. This gives us a 4/12 or a 33.33% chance to draw into Squirrel Girl on Turn 1.
We can go ahead and do this for every turn, and even go all the way up to 12 draws, in case we have other card draw effects. We will start with 4 draws because you always have 4 cards at the start of a game. That looks like this:
There are a few interesting takeaways from this. One of the biggest ones is that you have a 75% chance to see a specific card after 9 draws, or Turn 6. Because the deck size in Marvel Snap is so small, you are able to more easily build around a card, knowing you will see it in your hand approximately 3 out of every 4 games.
You can also see the impact that drawing even 1 additional card can have on a deck, whether it is through a location like Nova Roma (175) or a card's ability like Adam Warlock (201). As players unlock more cards, it will be interesting to see if card draw cards can have a positive impact on decks.
These statistics are very interesting, but what if we want to do something more advanced? For example, what if in addition to Squirrel Girl, we also have Ant-Man (7) in our deck, and we want to know the odds of having at least 1 of those 2 cards in our opening hand? Knowing these numbers would help us analyze and improve our curve. In order to do this math, we will need to use something called hypergeometric probability.
Hypergeometric Probability
Simply put, hypergeometric probability is the chance for an event to occur at least "x" number of times with "y" number of draws without replacement. While this has many applications, we are mainly interested in how it helps us improve our Marvel Snap deck and gameplay.
I won't explain too much about the math side of things, as that would fill up the article very quickly, but if you are interested in learning more about this, check out this Wikipedia article that goes over hypergeometric probability and its applications.
Let's try and better understand the concept of hypergeometric probability by using the Squirrel Girl and Ant-Man scenario. We want to have at least 1 of those 2 cards in our hand at the start of a game. So there are 4 possible events that can occur in this scenario:
Event 1: We have only Squirrel Girl in our opening hand.
Event 2: We have only Ant-Man in our opening hand.
Event 3: We have both Squirrel Girl and Ant-Man in our opening hand.
Event 4: We have neither Squirrel Girl nor Ant-Man in our opening hand.
When we use hypergeometric probability in this case, we want to know how often we draw at least 1 of these cards. For this example, it means that if Event 1, 2, or 3 occurs, it satisfies our conditions. We can calculate the odds of 1 of those events occurring using hypergeometric probability. Don't worry, we won't take the time to do this by hand. We can use a calculator to get this done much faster. We end up with a 57.57% chance to have at least 1 of those 2 cards in our opening hand. Much better than 33.33% for only running Squirrel Girl!
Now, what if we want to know this information for 6 1-cost cards, or what if you want to know how often you will find one of your 2 4-cost cards by Turn 4? I went ahead and made a chart that showcases this information for us, so you can reference it while deck-building!
We can make some interesting observations by looking at these statistics. Firstly, there are diminishing returns for including more cards of the same cost. Going from 1 1-cost card to 2 1-cost cards gives an additional 24.24% chance to have a 1-cost card on Turn 1 while going from 3 1-cost cards to 4 1-cost cards only gives an additional 11.04%. So it's important to weigh the value of an increased chance to have a certain cost card by a certain card versus the deck slot that that card takes up.
Another very interesting thing to look at is that decks generally don't need many big finisher cards to win games unless you are playing a deck built around cards like Wave (139). If you are running just 2 finisher cards, you have a 95.46% chance to find at least one of them by Turn 6, which is very good.
Be sure to keep this information in mind while deck-building! It can help to know how frequently you will draw the cards you need.
Conclusion
Managing probability in card games is a very important skill, and knowing how your deckbuilding decisions impact card draw probability lets you improve your skills in order to win more games. Learning more about how you can use the mathematical side of the game to your advantage will help you in the long run!
That's all for this article! There are still quite a few questions regarding the probability that we have not answered. What if you want to know how often you will have a certain 2 or 3 card combo? How do cards like America Chavez (4) impact your draws? We will take a look at these questions and more in future articles! Thanks for reading!
