What a rarity tier is
Collectible card games usually sort cards into tiers such as common, uncommon, rare and something rarer still. Each pack contains a mix, and the tiers differ in how often they appear. The numbers below are invented examples for illustration, not the rates of any real product. Always check what a publisher actually states, and remember that not every game publishes its rates in detail.
From a rate to an expected count
If a game says that a certain rarity appears once per pack on average, then ten packs contain about ten cards of that rarity. If it says one in eight packs contains a particular top tier card, then the expected number per pack is one eighth, or 0.125. Over ten packs, expect 1.25.
This expected value is just the rate times the number of packs. It describes the long-run average, which can differ from what happens in any particular box.
Chance of at least one
Suppose the top tier appears in one in eight packs, independently, so that each pack has a 7 in 8 chance of not containing it. The chance of finding none in ten packs is (7/8) raised to the tenth power, about 0.263. So the chance of at least one in ten packs is about 0.737, or 73.7 percent.
Even though the expected count in ten packs is 1.25, you will open no top-tier card at all in roughly one set of ten out of four. After eight packs, which is the number the rate suggests as “enough”, the chance of none is about 34 percent. A rate of one in eight does not mean one in every eight, and this is a common misreading. It is the same reasoning that applies to any probability of at least one success.
A specific card is much harder
Most buyers do not want any top-tier card; they want a particular one. Suppose there are 20 different top-tier cards and each is equally likely when a top-tier card appears. Then the chance that a given pack contains your specific card is (1/8) times (1/20) = 1/160.
- The expected number of packs for one copy is 160.
- After 160 packs, the chance of having found it is 1 minus (159/160) raised to the 160th power, about 63 percent. So even after buying the “expected” number, around 37 percent of the time you are still without it.
- The point at which it becomes more likely than not is around 111 packs.
- After 320 packs, the chance reaches about 87 percent.
These calculations are for a single specific card. This kind of waiting time is described by the geometric distribution, and the long, uneven tail is its defining feature: most people find it earlier than the average, while a smaller number wait much longer.
Slots, not just percentages
Packs are often built from slots rather than drawing every card from the same pool. A pack might have several commons, one slot that is usually uncommon or better, and a slot reserved for rare or better. The advertised rate for a tier is then a combination of the slot structure and the chance within each slot. For this reason, a rule such as “one rare per pack” can coexist with a very low rate for the top tier. Reading the full pack structure matters more than a headline number.
Another detail is that some products contain cards that are not drawn independently. Printing and sorting processes can mean that cards are not distributed as a pure random draw from an infinite pool, and some games include features designed to limit duplicates within a pack. When this applies, the independent calculation above is an approximation, and the real chances may differ somewhat.
Weighting by value
Different cards have different market values, so it is natural to ask what a pack is “worth”. The method is a weighted average: multiply the chance of each outcome by its value, and add the results. The idea of weighted averages is the basis of fair price estimates for random rewards.
This calculation has two weaknesses. Card values change, sometimes quickly, and the most valuable cards can dominate the average even though most packs contain none of them. Because the average is lifted by rare high-value cards, the median outcome, what a typical pack is worth, is usually lower than the average. A pack that is worth a certain amount on average may be worth far less in most openings and far more in a few.
Expected value and the long run
Opening many packs makes your results closer to the average, a pattern described by the law of large numbers. It does not make any single pack or box more predictable, and a dry streak does not mean that a pull is “due”. Each independent pack has the same chance as the last one, which is the content of the gamblers fallacy warning. Some products and digital games do add a guarantee after a number of unsuccessful openings, a mechanism known as a pity timer, and if one exists, it changes the calculation. The published guarantee would need to be included.
Practical points
- Treat published rates as estimates of long-run frequency. If a publisher does not state the rates, you cannot calculate the expected contents with confidence.
- Work out the chance of getting what you want by a given number of packs, not just the average.
- Compare the cost of buying packs with the cost of buying the single card you want. For a specific card, buying it directly is often more predictable, though prices vary and the comparison depends on the market.
- Be cautious with spending. Opening packs is a game of chance with no way to improve your odds, and the outcomes are uncertain even when the rates are known. Set a budget before you start and stick to it.
The mathematics here is simple, but the conclusion is worth stating clearly: published rates tell you about the average, and they say little about how long you may have to wait.