Photocards · PocaBinder
How many albums does it take to collect every photocard in a set
With one random photocard per album and every card equally likely, a set of n cards takes n × (1 + 1/2 + … + 1/n) albums on average. That is about 11 albums for 5 cards, 18 for 7, 25 for 9, and 37 for 12. The last missing card alone takes n albums on average.
Why a set costs more than its card count
Many K-pop albums come with a photocard packed at random from a larger set, so buying one album does not tell you which member you will get, as K-Pop Atlas explains in its guide to album contents. Reporting on an investigation by Korea's Fair Trade Commission in 2023, Digital Music News described the result: because the cards are placed randomly, the most dedicated fans buy several copies of the same album to raise their chances of getting the card they want.
How many copies is a probability question with a well-known answer. Mathematicians call it the coupon collector's problem: if every box contains one of n kinds of coupon, how many boxes do you need to buy to collect all of them? Swap boxes for albums and coupons for photocards and you have the collector's version.
The formula, step by step
Wikipedia's entry on the coupon collector's problem gives the expected number of draws as n times the n-th harmonic number: n × (1 + 1/2 + 1/3 + … + 1/n). Each term is the wait for one more new card. Once you hold i − 1 different cards, the chance that the next album brings a new one is (n − i + 1)/n, and the wait for it averages n/(n − i + 1) albums.
Take a seven-card set. The first album is always new, so that wait is 1. The second new card takes 7/6 albums on average, then 7/5, 7/4, 7/3, 7/2, and finally 7 for the last one. Added up, that is about 18.2 albums.
Bigger pools climb fast. The same Wikipedia article notes that collecting 50 coupons takes about 225 trials on average.
Expected albums by set size
These are n × (1 + 1/2 + … + 1/n), rounded, for one random card per album and an even pool. 4 cards: about 8.3 albums. 5 cards: 11.4. 6 cards: 14.7. 7 cards: 18.2. 8 cards: 21.7.
9 cards: about 25.5 albums. 10 cards: 29.3. 12 cards: 37.2. 13 cards: 41.3. 14 cards: 45.5.
The n that goes into the formula is the number of different cards that can come out of that album, not the number of members. If each of seven members has two designs in the same pool, n is 14 and the average is about 45.5 albums, not 18.
Why the last few cards cost the most
The waits are lopsided. In a nine-card set, the first five different cards take about 6.7 albums together on average. The last four take about 18.8: 9/4, then 9/3, then 9/2, then 9 for the final card. Nearly three quarters of the expected total goes to those last four.
That is why a set that felt almost finished can stall. With one card left in a nine-card pool, every album has a 1 in 9 chance of being the one, and the wait averages nine more albums no matter how many you have already opened. The earlier albums do not make the next one luckier.
When the pool is not even
Every number above assumes each card is equally likely, and real print runs are not guaranteed to be even. Digital Music News reported that the Fair Trade Commission's 2023 investigation looked into suspicions that companies had altered production numbers to make some photocards rarer within a set. The report describes what was being investigated; it does not tell you how any particular album is packed.
If one card turns up less often, the wait for that card alone averages 1 divided by its chance per album, by the same reasoning as the formula above. A card with a 1 in 18 chance per album takes 18 albums on average by itself, whatever happens with the rest of the set.
Pre-order benefit cards sit outside this math. K-Pop Atlas notes that a POB is separate from the inclusions sealed inside the album, so opening more albums does not draw from the POB set.
Common mistakes when planning a set
Planning for n albums. Buying as many albums as there are cards rarely completes a set. For a five-card set, the chance that five albums give five different cards is 5!/5^5, which is under 4 percent.
Counting members instead of cards. Use the number of distinct cards in the pool, and check whether each album version has its own set of cards or shares one with the others.
Treating the average as a promise. The formula gives an average over many attempts. Some collectors finish early; others need far more albums, mostly for the last card.
Losing track of what is missing. The math only helps when you know n and how many you already hold. PocaBinder makes a slot for every member and version you enter and shows a count such as 5 of 9 collected, so the number still missing is always in front of you.
Frequently asked questions
Does buying several albums at once change the odds?
Not in the coupon collector model. Each album is an independent draw from the same pool, so ten albums bought together behave the same as ten bought one at a time.
What if each album has two random photocards?
If both cards are independent draws from the same pool, you need about the same number of draws, so the album count roughly halves. For a seven-card set that is about 18 draws, or around 9 albums.
Are pre-order benefit photocards part of the random set?
No. K-Pop Atlas describes a POB as an extra item offered by a specific retailer or during a particular order window, separate from what is sealed inside the album.
Is the expected number a guarantee?
No. It is an average over many attempts. A single collector can finish sooner or need many more albums, and the last card is where most of the variation comes from.