Card Draw Math in Star Wars: Unlimited
How likely are you to draw a given card, and how can you do the math to figure it out yourself?
This article is partly in response to a discussion I saw on Discord earlier this week, and partly an article I’ve wanted to put out for a while. The point is to show you the math behind drawing cards in Unlimited, and how that compares to some other games out there. This can be useful in both gameplay and deckbuilding.
For Reference
Here are a few important probabilities that you can read before we get into depth.
The probability of drawing a card in your opening hand with no mulligan if you run x copies is:
3 - 32%, 2 - 23%, 1 - 12%
With Mulligan:
3 - 54%, 2 - 41%, 1 - 23%
The probability of seeing a card by turn 3 with no mulligan if you run x copies is:
3 - 49%, 2 - 36%, 1- 20%
With Mulligan:
3 - 66%, 2 - 52%, 1 - 30%
The Underlying Math
When drawing cards from a deck, at a fundamental level you are sampling randomly without replacement from a population with a finite size. (If that makes no sense to you, drawing from a deck of cards is often used to explain this concept in stats classes). Mathematically, we can model this using what’s called a hypergeometric distribution. Here’s a link to a calculator that you can do all the math in yourself, and here’s what it looks like:
The top 4 fields are what we input, and grayed out fields are the output. The only one that really matters to us is is the bottom one, so I crossed out the other ones. Here’s what all the parameters mean in this context:
Population size is just the initial number of cards in the deck. If you were calculating the odds of opening a given card, this would be 50. If you were calculating for a different round, it would be lower.
Number of Successes in Population is how ever many copies of the desired card you’re running. If you wanted to know the odds of drawing exactly Cloud City Wing Guard, and you’re running 3, you should put 3. Say you’re also running 3 copies of Cell Block Guard and you want to know the odds of drawing a sentinel unit, you would put 6.
Sample size is how many cards you’re drawing. So for opening hands it would be 6, if you wanted to find the odds of drawing a card by a given round, you would but 6 + 2 for every additional round(because you draw 2 cards per turn in Unlimited). For example, If you wanted to know the odds of drawing a card by round 3 you would put 10.
For number of successes in sample just put 1 unless you wanted to know the probability of opening multiples of a card, or if you’re doing something like calculating the probability of getting max value out of Tarkin. The number you put here becomes the lowercase x in the output fields.
The output fields are the probability that X(the number of successes) is some way in relation to the number you put into the last field. The last one is the probability that your number of successes is greater than or equal to what you input into the last field. In real terms, this means the odds of getting at least one copy of a card in your sample. The other output fields might be fun to look at, but the last one is usually what really matters. Don’t get tripped up by the first field (X=x) because that is the probability that you draw exactly one copy of a card, not the probability that you draw it at all.
Here’s a real example. In this case, we want to find the odds of drawing a card that we run 3 copies of in our opening hand.
The value we get here in the bottom field is about .32, or 32%. This means we have a 32% chance of drawing at least one copy of a card that we run 3 copies of in our opening hand. You can check above for a list of these probabilities for other numbers of cards, and now you know how to check my math.
You can know use this knowledge to find the probability of drawing a given card in almost any scenario, given that you know how many cards are in your deck, how many copies of the desired card or group of cards are, and how many cards you will be drawing/how many turns you will be taking.
Taking Mulligans Into Account
It might be tempting to account for mulligans by writing 12 instead of 6 in the sample size, but this is WRONG because in Unlimited, you replace the cards you mulligan before drawing them again. To account for mulligans, we need to use Unions. The math here is pretty simple and be can be summed up with the following rule: the probability of success is 1 minus the probability of failure.
Let’s take our example from above. We have a 32% chance of opening a 3 of. If we have a mulligan, we’re allowed to take that chance twice. The union of these two events is the probability that we succeed on either one. So we use the above rule. The probability we don’t draw our card on the first try is:
1-.32=.68
So the probability of failing twice is:
.68*.68=.4624, or about .46
Thus the probability of succeeding at least once is
1-.46=.54
So we can say we have a 54% of drawing a card we run 3 copies of in our opening hand if we take a mulligan. Keep in mind this is inclusive of the world where we get it before mulliganing, we don’t assume we will mulligan if we draw it in our first six. This also doesn’t mean that you have a 54% chance of drawing it the second time, that probability is still 32%. So if you whiff on your first six, don’t think you still have a coin flip of drawing the card if you mulligan for it. The .54 is useful only if we don’t know the outcome of the first draw(look up Bayes theorem if you want to know more).
You can use this calculator to do your math for you if you like. Input your probabilities like this:
Press calculate, and check for your answer here:
Mulligans and Future Rounds
We can use a combination of both of these calculators to figure out the probability of drawing a card by a certain round taking mulligans into account. To do this, find the probability of drawing a certain card by the nth round after your opening hand. Use the hypergeometric calculator for this, making sure to change the population size to 44(50-the 6 cards in your opening hand) and 2*the number of rounds beyond the first. Above, I calculated the odds of drawing a card by round three, so I used 4, 2 for round two and 2 for round three. Then input this probability and the muliganized probability we just learned how to find into the Union calculator, and there’s you answer.
There’s a list of these probabilities calculated for round 3 above if you want to check my math/check your math if you have enough faith in me.
Unlimited vs. Other Games
How does Unlimited compare to other games in terms of these probabilities? For this exercise, I’m going to use Star Wars: Destiny and Disney: Lorcana as examples. Both have different deck sizes and different mulligan structures, so they can offer good points of comparison.
In Destiny, we have 30 card decks with maximum 2 copies of a given card. We have opening hands of 5 cards, and we can choose to mulligan any number of them. We still replace them when we mulligan. Here’s a table for the odds of drawing a card with any given number of copies when we mulligan a given number of cards:
The second line is really the most important to us, because you could run 2 copies of a given card. This probability ranges from 31%-52%, which is actually slightly below the probability in Unlimited(32% or 54%), although by a very very small amount. However, Destiny allows for a much safer way of increasing your odds by mulliganing only a select number of cards. It’s also worth noting that you draw up to 5 cards every round in Destiny, so the probability of seeing cards by a given future round is much higher in Destiny, although rounds are significantly longer in Destiny as well.
In Lorcana, we have 60 card decks, 4 copies of a card, 7 card hands, and for mulligans we can put back any number of cards(like Destiny) but we don’t replace them first. This means we can just add the number of cards we mulligan to the sample size in the hypergeometric calculator. Here’s a table that I made using that method.
If we look at the top row, we can see the probability ranges from 39%-67% of opening a given card if you run the maximum number of copies allowed, which is a good bit higher than the probability in Unlimited.
Conclusions
Unlimited is a somewhat middle of the road game when it comes the odds of drawing your cards. If you mulligan, you have better than coin flip odds of opening a card you run the maximum number of copies of. You also have good odds of seeing cards eventually if you don’t open them because you’re drawing two new cards every round.
You should also now be able to figure out these probabilities yourself if you ever need them to help build a deck.
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