Pack Odds 101
- Understanding Probabilities and Print Runs in Trading Card Collecting
For those immersed in the world of trading cards, a common question emerges: What is the rarity of a particular card? Rarity is twofold: there’s the fundamental rarity, reflecting the actual quantity of cards in existence, and the perceived rarity, which hinges on the demand relative to the supply. It’s the latter that predominantly influences a card’s value.
While perceived rarity is a function of fundamental rarity, it is intertwined with numerous complex variables beyond the scope of this article. On the other hand, fundamental rarity is a realm we can more accurately navigate.
For serial numbered cards, determining their quantity is straightforward since the number is per defintion already imprinted on each card. However, the distinction between, say, 20 parallels each numberd to 25, differing only in color, versus a single uniquely designed insert set also numbered to 25, makes a huge difference. Yet, delving into this discussion also exceeds the confines of this article.
The intricacy of rarity further deepens when considering inserts lacking serial numbering, a common scenario in cards issued before 2000. Tipically, for those cards pack odds were disclosed, indicating the likelihood of finding a specific set in every x packs. In the past, these odds were printed on packs and boxes; today, they can be sourced from online databases like Beckett.com.
In addition to odds, one must ascertain the number of cards within a given insert set to calculate the probability of obtaining a particular card from a pack. Armed with this probability, one can deduce the number of packs needed to secure at least one desired card. The simplest approach is to compute the counter-probability of not obtaining any copies of the desired card when opening a specified number of packs.
Let q represent the probability of obtaining a specific card in a pack (1 divided by the product of pack odds and number of different cards in a set), n the number of packs you must open, and P the probability of pulling at least one copy of that specific card when opening n packs. The counter-probability Q of not pulling any copies of a specific card with n packs is calculated as follows:
Thus, the probability we are looking for denotes:
With the probability q and the number of packs n provided, you can readily calculate P using Eq. 1. Typically, our focus lies in determining the number of packs required to pull at least one copy of a specific card with a given probability. Hence, we need to solve Eq. 1 for n:
Eq. 2 provides a pathway to eventually address the query of how many packs one needs to open to procure at least one copy of a specific card with the probability of P. I’ve opted for the natural logarithm with base e, but you can employ any logarithm provided consistency between the numerator and denominator. It’s worth noting that this calculation assumes a continuous printing process for a given series. In reality, there exists a finite number of packs produced with a finite total number of cards in a given insert set. While Eq. 2 implies that you can never attain a probability of 100% for obtaining a particular card, this isn’t true in practice, as you’ll inevitably obtain the card when opening all produced packs. However, if we knew the initial quantity of cards produced, determining the scarcity of any given card would be straightforward. Bearing this in mind, Eq. 2 functions adequately for a sufficiently large number of produced packs.
To provide an example, let’s consider one of my favorite cards: the 2001–02 Fleer Marquee “We’re Number One” featuring Shaquille O’Neal, as shown in Pic. 1. This insert set was randomly distributed in packs at a rate of one in 240, comprising eleven cards. The probability p of pulling the card in a single pack is calculated as follows:
By applying Eq. 2, we can ascertain the number of packs required to be opened in order to obtain at least one copy of Shaq’s “We’re Number One” insert, with different probabilities denoted as P:
From Table 1, it’s evident that nearly 2,000 packs need to be opened to acquire a copy of the card with a probability exceeding 50%. The number of packs required theoretically increases tenfold when aiming for a probability of 99.9%. For this particular product, each box contained 20 packs, implying that approximately 1,000 boxes would need to be opened to achieve a 99.9% likelihood.
Another motive for selecting this specific card for illustration is the potential to estimate the total number of packs produced for the corresponding product line, 2001–02 Fleer Marquee. Each box contained one serial numbered jumbo box-topper card. There were four variations of these oversized cards: The primary Feature Presentation Film set, which comprised 14 players and displayed a single slide from an actual game film, numbered to 350; the 14-card Feature Presentation Triples parallel set, numbered to 100; the 10-card Feature Presentation Film/Jerseys set, numbered to 250; and one Vince Carter autographed version, numbered to 208. If we assume that each box indeed contained one of these numbered cards, it results in a maximum of 9,008 total boxes, assuming full insertion without any holdbacks. With each box containing 20 packs, this totals 180,160 packs produced.
With the total number of packs determined, we can estimate the actual print run of our card. Knowing that it was inserted once in every 240 packs and considering there are eleven players in the set, this equates to approximately 68 copies for each player under the assumption of an equal distribution of copies per player.
As a concluding fun exercise, let’s calculate the height of cards stacked atop each other after opening the required number of packs to obtain at least one of the Shaq “We’re Number One” inserts with a given probability. Assuming each card measures 0.4 mm and considering that a pack contains five cards, the heights for different minimal probabilities P are illustrated in Diagram 1.
As observed, to attain at least one of those cards with a modest probability of 50%, you accumulate a card stack towering at approximately four meters.
In summary, this article derived the formula for calculating the minimum probability of pulling a specific card within a given number of packs, or equivalently, the number of packs necessary to obtain at least one copy of a specific card. Despite this equation not considering the finite production of packs, it remains a useful approximation for scenarios involving a large number of produced packs. Furthermore, we discovered that if a product line features a set for which we know both the print run and the pulling odds, it’s possible to determine the total product volume. From there, we can derive the print run of any other set for which only the odds are given.
