Box Math: The Expected Value of Phantom Rage

There are many ways to enjoy Yu-Gi-Oh! and opening booster boxes is certainly one of them. There's a reason box openings form the most popular content segment on social media. However, in this article, we will leave personal reasons aside and focus on the cold hard math of opening booster boxes. Grab a tea and prepare for a short lesson in mathematics.

The Expected Value

The expected value, also known as weighted average, is a term used in probability theory and finds key applications in economics, finance, and many other subjects. It's a generalization of the average and takes into account that some outcomes are more likely than others. In our case: We cannot simply add up the value of all cards that could be in our booster box as some cards are very common and others are very rare. Pun totally intended.

The mathematical formula to calculate the expected value is fairly simple. We need information about two variables: the value of all cards in the set and their pull rates.

Determining the value of cards is the easiest part as Cardmarket always provides the most up-to-date reference point. As you may have guessed from the title, we will be using Phantom Rage as our case study as it is a recent core set that had enough time to stabilize its prices. You can simply use the product page of the particular expansion and apply the filters you would also use to price your cards.

To determine the pull rates, we will assume perfect randomization. Konami has made this part easier in recent years as we are more or less "guaranteed" to get two secret rare cards and four ultra rare cards per booster box. Starlight rare cards are included in every 24th booster box. We can filter through the singles overview to determine the number of cards of each rarity.

Let's start with the chase cards, most notably and .

There are 0.5 starlight rare cards per case and five different starlight rares in Phantom Rage, so the pull rate of a particular starlight rare is 0.1 per case. Remember that one case equals twelve booster boxes. We build the sum of all card values and multiply that sum with our pull rate per case.

Rinse and repeat with secret rares: there are 24 secret rares per case and ten different secret rares, so the pull rate of a particular secret rare is 2.4 per case. Once again we add up all the card prices and multiply that number with our pull rate.

One last ride for our ultra rares: there are 48 ultra rares per case on average and fourteen different ultra rares per set, so the calculated pull rate is roughly 3.4 of each per case. We add up all of their prices and multiply that number with our pull rate to receive an expected value of roughly €140 in ultra rares.

For super rares and commons we assume a bulk price of ten and two cents each, respectively, which roughly adds €67 to the value of our case.

Finally, we add up all of our results and conclude that the total expected value of twelve booster boxes is roughly €620—almost exactly the value of twelve Phantom Rage booster boxes!

Conclusions and Other Factors

First of all, we should note that the value of the cards in the box almost exactly equal the value of the sealed box. The law of large numbers states that the risks and rewards of opening high quantities of booster boxes equal each other out as long as the prices for both the sealed box and the cards are in balance.

Maybe more surprisingly our calculations tell us that on average, the ultra rares in a case are worth more than the starlight rares. If you want to minimize your risks, take a look at high value ultra rares such a and before getting all hyped about the starlight rares.

Last but not least, be aware that these numbers reflect averages and do not take variance into account. While the results will be true for large quantities, they do not provide enough information about buying one individual box. A starlight rare will almost guarantee you the best bang for your buck, but it is almost impossible to make up for the lack of a starlight rare in twelve, 24, or even 36 booster boxes.

What's your experience with or opinion on the matter? Please leave a comment below—especially if you would like to read more about the financial aspect of our favorite card game in the future.

Stay safe and may blessed be your pulls!
Tayfun

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Opinions expressed in this article are those of the author and not necessarily Cardmarket.