Overview
Trading card games such as Magic the Gathering, Yu-Gi-Oh and Pokemon are comprised of cards that are each assigned a rarity such as common, uncommon and rare (amongst other gameplay attributes). Players can purchase booster packs to expand their collection, which are sealed, randomised packages of cards, most of which contain cards worth less than the price of the booster pack, while some contain a chase card worth significantly more. Serious collectors sometimes consider the expected value of a booster box (which contains 10-36 booster packs, depending on the product) to determine if it is worth purchasing, using websites such as MTG Stocks. As someone who enjoys collecting and playing with trading cards instead of just selling them for profit, I am more interested in answering questions like:
- if I open $B$ booster packs, what is the probability of opening at least one of the cards I want?
- how many booster packs do I need to open to be more than 50% sure I will get at least one of the cards I want?
In this article, I define a generalised mathematical model of a TCG booster pack (which can be applied to any particular trading card game), and use it to answer the above questions. At the end of the article, you'll find an interactive calculator that you can use to input a booster pack model and compute answers to these questions. This calculator was implemented in JavaScript (using jQuery), and you can read the implementation by viewing the source code of this webpage (typically available in the right-click menu).
Mathematical model
Let $\mathcal{C}$ be the set of all cards we could possibly find in the booster pack, and let $\mathcal{R}$ be the set of all card rarities present in $\mathcal{C}$. For a given rarity $r \in \mathcal{R}$, let $S_r$ be number of cards of that rarity in $\mathcal{C}$, such that $$\sum_{r \in \mathcal{R}} S_r \> = \> |\mathcal{C}|.$$ It is important that we count variants of the same card (e.g., a foil version) as separate cards in $\mathcal{C}$ if they have different likelihoods of being found in the booster pack. Imagine a hypothetical TCG booster pack whose card set $\mathcal{C}$ contains the rarities:
- common with $S_C = 100$,
- uncommon with $S_U = 75$,
- rare with $S_R = 50$,
- mythic rare with $S_M = 25$ and
- foil variants of each of the above rarities, with $S_{FC} = S_C$, $S_{FU} = S_U$, $S_{FR} = S_R$ and $S_{FM} = S_M$.
Let $N$ be the number of cards in a booster pack, and for any $n \in \{1, \ldots, N\}$ and $r \in \mathcal{R}$, let $p_n^r$ denote the probability that card $n$ in the pack is of rarity $r$. If we assume that there are no special rules about booster box guarantees or duplicate cards in packs (i.e., each booster pack is independent and identically distributed, and within a booster pack, each card slot is sampled independently), then the $N \times |\mathcal{R}|$ matrix $\boldsymbol{P}$ of these probabilities (along with the rarity set counts $S_r$) uniquely defines the booster pack. These probabilities can typically be found on a trading card game's website - for example, the image below shows an explanation of a booster pack from a recent TCG called Sorcery: Contested Realm.
Continuing our hypothetical example, let's say that the booster pack contains 12 cards and is laid out as follows:
- cards 1-7: always common, so for $n \in \{ 1, \ldots, 7 \}$, $p_N^C = 1$ and $p_n^r = 0$ for all $r \neq C$;
- cards 8-10: always uncommon, so for $n \in \{ 8, 9, 10 \}$, $p_N^U = 1$ and $p_n^r = 0$ for all $r \neq U$;
- card 11: 80% likely to be rare and 20% likely to be mythic rare, so $p_{11}^R = 0.8$, $p_{11}^M = 0.2$ and $p_{11}^C = p_{11}^U = 0$; and
- card 12: 50% foil common, 30% foil uncommon, 15% foil rare and 5% foil mythic, so $p_{12}^{FC} = 0.5$, $p_{12}^{FU} = 0.3$, $p_{12}^{FR} = 0.15$ and $p_{12}^{FM} = 0.05$.
Notice that $$\sum_{r \in \mathcal{R}} p_n^r \> = \> 1,$$ for all $n$, meaning each row of $\boldsymbol{P}$ sums to 1, because each card slot is guaranteed to contain a card.
Likelihood of opening a desired card from multiple booster packs
Let's say we have $B \in \mathbb{N}$ booster packs to open, and we have identified a subset of desired cards $\mathcal{D} \subset \mathcal{C}$ that we would like to find, and that we would be happy to open at least one of these cards. For each rarity $r \in \mathcal{R}$, let $0 \leq d_r \leq S_r$ be the number of cards in our desired subset at that rarity, so $$\sum_{r \in \mathcal{R}} d_r = |\mathcal{D}|.$$ Then $$\begin{align} \mathbb{P}(\text{open at least one desired card}) \> &= \> 1 - \mathbb{P}(\text{open no desired cards}) \\ &= \> 1 - \mathbb{P}(\text{open no desired cards from a booster pack})^B \\ &= \> 1 - \left( \prod_{n=1}^N \mathbb{P}(\text{card } n \text{ is not desired})\right)^B \\ &= \> 1 - \left( \prod_{n=1}^N \left( 1 - \mathbb{P}(\text{card } n \text{ is desired}) \right) \right)^B \\ &= \> 1 - \left( \prod_{n=1}^N \left( 1 - \sum_{r \in \mathcal{R}} \frac{d_r p_n^r}{S_r} \right) \right)^B, \end{align}$$ because the likelihood of opening a specific card of rarity $r \in \mathcal{R}$ from the $n^{\text{th}}$ slot of a booster pack is $\frac{p_n^r}{S_r}$, which you can verify by noticing that the sum of these likelihoods $$\sum_{r \in \mathcal{R}} \sum_{c = 1}^{S_r} \frac{p_n^r}{S_r} \> = \> \sum_{r \in \mathcal{R}} p_n^r \> = \> 1.$$
Continuing the example, suppose that we have 8 booster packs to open and our pool of desired cards $\mathcal{D}$ consists of 4 rare cards and 8 mythic rare cards, which we would be happy to find as either non-foil or foil, so $d_R = d_{FR} = 4$, $d_M = d_{FM} = 8$ and $d_C = d_{FC} = d_U = d_{FU} = 0$. Then $$\begin{align} \mathbb{P}(\text{open at least one desired card}) \> &= \> 1 - \left( \prod_{n=1}^{12} \left( 1 - \sum_{r \in \mathcal{R}} \frac{d_r p_n^r}{S_r} \right) \right)^8 \\ &= \> 1 - \left( \left( 1 - \frac{d_R p_{11}^R}{S_R} - \frac{d_M p_{11}^M}{S_M} \right) \left( 1 - \frac{d_{FR} p_{12}^{FR}}{S_{FR}} - \frac{d_{FM} p_{12}^{FM}}{S_{FM}} \right) \right)^8 \\ &= \> 1 - \left( \left( 1 - \frac{4 \times 0.8}{50} - \frac{8 \times 0.2}{25} \right) \left( 1 - \frac{4 \times 0.15}{50} - \frac{8 \times 0.05}{25} \right) \right)^8 \\ &= \> 1 - \left( \left( 1 - 0.064 - 0.064 \right) \left( 1 - 0.012 - 0.016 \right) \right)^8 \\ &= \> 1 - ( 0.872 \times 0.972 )^8 \\ &\approx \> 0.73, \end{align}$$ so we are 73% likely to open a desired card and be satisfied with our purchase.
Number of booster packs required to be confident in opening a desired card
Let $0 < x < 1$ be the confidence we want to have in opening at least one desired card from $\mathcal{D} \subset \mathcal{C}$, and we wish to find the minimum number of booster packs $B$ we need to open to achieve this confidence. In the previous section, we showed that $$\mathbb{P}(\text{open at least one desired card}) \> = \> 1 - \left( \prod_{n=1}^N \left( 1 - \sum_{r \in \mathcal{R}} \frac{d_r p_n^r}{S_r} \right) \right)^B.$$ Substituting $x$, we have that $$x \> \leq \> 1 - \left( \prod_{n=1}^N \left( 1 - \sum_{r \in \mathcal{R}} \frac{d_r p_n^r}{S_r} \right) \right)^B,$$ and hence $$B \> \geq \> \frac{\log(1-x)}{\log\left( \prod_{n=1}^N \left( 1 - \sum_{r \in \mathcal{R}} \frac{d_r p_n^r}{S_r} \right) \right) }.$$
If we take the desired subset $\mathcal{D}$ containing 4 rare and 8 mythic rare cards from the previous example, and let $x = 0.5$ for a target 50% confidence in opening a desired card, we have that $$B \> \geq \> \frac{\log(0.5)}{\log(0.847584)} \> \approx \> 4.19,$$ so rounding up, we need to open 5 booster packs to be 50% confident that we'll open a desired card.
Interactive booster pack model
Below is an interactive table which you can use to define a booster pack model for calculating answers to the questions discussed in this article. The table is pre-populated with the hypothetical booster pack used in the examples above, but you can add & remove card slots and rarities, adjust the likelihoods & set sizes and rename the rarities to fit this model to a real TCG booster pack you would like to query. The bottom row of the table is for inputting the desired subset of cards $\mathcal{D}$. Note that leaving a field blank is equivalent to setting it to 0. After you're finished defining your model, you can use the "Check model" button to check if you have defined a valid booster pack. If you reload the page, the model will be reset to the default hypothetical example.
Below the table, enter a number of booster packs to compute the likelihood of opening at least one desired card, or enter a confidence threshold to compute the number of booster packs required to achieve that confidence.
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booster packs = likelihood
References
- Sison, Michael. "Money and Mathematics: Booster Packs." SixPrizes, 10 Apr. 2011, sixprizes.com/2011/04/10/money-and-mathematics-booster/.
If you have any corrections or questions - please send me an email and we can have a discussion. Any edits or additions to the original article & code will be noted here.