Pack Odds 101
By: Shaq Card Collector De
Tags: Probability, Trading Cards, Shaquille ONeal, Basketball Cards

- Deriving the Probability of Pulling a specific Card

For everyone who collects trading cards, the following question arises: How rare is a given card? The rarity of a card manifests in two ways: The fundamental rarity, i.e. how many cards are actually in existence and the perceived rarity, i.e. how big is the demand in relation to the supply. The latter attribute is, after all, the major driver for a card’s value. Even though the perceived rarity is a function of the fundamental rarity, it entails a lot more variables that are rather complex to determine and far beyond the scope of my expertise. The fundamental rarity, on the other hand, is a question which we can solve much more accurately. For serial numbered cards it is trivial to know how many cards are in existence as the number is printed on the particular card. Now you could argue that it makes quite a difference on whether there are 20 different parallels, only differing by coloring, numbered to 25, or just one uniquely designed insert set numbered to 25. But this topic is also beyond the scope of this article.

The topic of rarity becomes a little more nuanced when you consider inserts that don’t feature a serial number as was the case for most of the inserts from the 00s and earlier. For those cards usually pack odds were given, i.e. it was published that a certain set was randomly inserted in every x packs. Pack odds for the various insert and parallel sets used to be printed on the packs and boxes. Nowadays, they can still be found on the Beckett database.

Besides the odds, you also need the number of cards in a given insert set to determine the probability to pull a specific card from a pack. With this probability, you can determine the number of packs you need to open to receive at least one of those cards. The easiest way to find that number of packs is to calculate the counter-probability to not get any copies of the card you desire when opening a given number of packs.

Let p be the probability to obtain a specific card in a pack, n the number of packs you (have to), and P the probability with which you pull at least one copy of that specific card. The counter-probability Q to pull no copy of a specific card with n packs is given as:

Thus, the probability we are looking for denotes:

Eq. 1

Given the odds as well as the number of packs you can easily calculate P from Eq. 1. Usually, we are interested in the number of packs needed to obtain at least one copy of a specific card with a given probability. Therefore, we need to solve Eq. 1 for n:

Eq. 2

Eq. 2 allows for eventually answering the question of how many packs you need to open to pull at least one copy of a specific card with the probability of P. I chose the natural logarithm with base e but of course you can use any logarithm as long as you use the same one in both the numerator and denominator. Consider that this calculation assumes that the printing of a given series is a continuous process. In reality there is only a finite number of packs produced with a finite number of total cards of a given insert set. While according to Eq. 2 you will never be able to obtain a given card with a probability of 100% in reality this is not true as you will certainly pull the card when you open all packs produced. But if we knew the amount of cards produced in the first place, we would also know the scarcity of any given card in a rather trivial way. Keeping that in mind, Eq. 2 works fine for a sufficiently large number of produced packs.

Pic. 1

For illustrative purposes let’s look at an example. For some reason, one of my favorite cards is the 01–02 Fleer Marquee We’re Number One of Shaquille O’Neal shown in Pic. 1. This insert set was randomly seeded in packs at the rate of one in 240 and consists of eleven cards. The probability q of pulling the card in a given pack is given by:

Applying Eq. 2, we can determine the number of packs we need to open to pull at least one copy of the We’re Number One insert of Shaq with various probabilities P:

Tab. 1

From Tab. 1, we find that you have to open almost 2,000 packs to obtain a copy of the card with a probability of more than 50%. That number of packs theoretically ten folds when you want to obtain a probability of 99.9%. In the case of that product, one box consisted of 20 packs meaning that you would have needed to open about 1,000 boxes for a likelihood of 99.9%.

Another reason I chose this particular card for my example is that we might be able to estimate the total number of packs produced for this product line. Each box contained a jumbo box-topper pack featuring one numbered oversized card. There were three different kinds of those cards: two fourteen-player sets numbered to 350 and 100, respectively, as well as one autographed version of one player numbered to 208. As each box should have contained exactly one of those numbered cards this results in a maximum number of 6,508 total boxes, assuming that each card was actually inserted and nothing was held back. With each box consisting of 20 packs there were 130,160 packs produced. Having determined thetotal number of packs, we are able to estimate the actual print run of our card. Since we know that it was inserted 1 in every 240 packs and that there are eleven players in the set this results in 49 copies for each player under the assumption that each player has an equal amount of copies.

Just for fun, we will derive the height of cards stacked atop each other when you have opened the number of packs necessary to pull at least one of the Shaq We’re Number One inserts with a given probability. Assuming a card measures 0.4 mm and given the fact that a pack contained five cards, the heights for the various minimal probabilities P is depicted in Diag. 1:

Diag. 1

As we can see, to pull at least one of those cards with a modest probability of 50% you amass a card stack towering at about four meters.

To summarize, in this article we derived the formula for calculating the minimum probability of pulling a specific card in a given number of packs or equivalently, the number of packs necessary to pull at least one copy of a specific card. Even though this equation doesn’t not take into account the fact that only a finite number of packs is produced it is still a good approximation for a large number produced packs and realistic probabilies.