COMICS / SF / FANTASY / COLLECTINGA 1 in N pull rate gives 1 minus (1 minus 1/N) to the power k, and a full box can still miss
If a hit comes 1 in N packs and each pack is an independent draw, the chance of at least one hit in k packs is 1 minus (1 minus 1/N) to the power k. At 1 in 21 and 10 packs that is 38.6%. At 1 in 21 and 36 packs it is 82.7%, which leaves 17.3% of openings with no hit at all. The same arithmetic runs in the pack odds calculator.
The formula needs one rate and one count
Take N from the stated odds and k from the packs opened.
- The chance one pack is a miss is 1 minus 1/N.
- The chance k packs are all misses is that number to the power k.
- The chance of at least one hit is 1 minus the all-miss chance.
- The expected number of hits is k divided by N. It is an average over many openings, and any one opening lands above or below it.
- The packs needed for a target chance p is the smallest whole k where the chance reaches p. In symbols, k is the smallest whole number at or above ln(1 minus p) divided by ln(1 minus 1/N).
The calculator takes the odds from you and holds none of its own. That keeps every number in it a number you supplied.
The probability table is arithmetic, not data
The grid below applies the formula to illustrative odds. The 1 in N values are examples chosen to span common ranges, and they do not describe any set. The 36 packs column stands in for a box and is also an example.
| Stated odds | 1 packs | 5 packs | 10 packs | 36 packs |
|---|---|---|---|---|
| 1 in 5 | 20.0% | 67.2% | 89.3% | 99.9+% |
| 1 in 10 | 10.0% | 41.0% | 65.1% | 97.7% |
| 1 in 21 | 4.8% | 21.6% | 38.6% | 82.7% |
| 1 in 50 | 2.0% | 9.6% | 18.3% | 51.7% |
| 1 in 100 | 1.0% | 4.9% | 9.6% | 30.4% |
Read across a row to see how fast the chance builds. At 1 in 10, five packs give 41.0% and ten packs give 65.1%. Ten packs at 1 in 10 still come up empty 34.9% of the time, about one opening in three. Read down a column to see how much each step in rarity costs: at 36 packs the chance drops from 99.9+% at 1 in 5 to 30.4% at 1 in 100.
Packs needed scales with N, from about 0.7 N for even odds to 3 N for 95 percent
Asking how many packs reach a target chance reverses the formula.
| Stated odds | Packs for 50% | Packs for 90% | Packs for 95% |
|---|---|---|---|
| 1 in 5 | 4 | 11 | 14 |
| 1 in 10 | 7 | 22 | 29 |
| 1 in 21 | 15 | 48 | 62 |
| 1 in 50 | 35 | 114 | 149 |
| 1 in 100 | 69 | 230 | 299 |
Reaching even odds takes fewer than N packs, about 0.69 times N. Reaching 90 percent takes about 2.3 times N, and 95 percent about 3 times N. A 1 in 100 hit takes 230 packs to reach 90 percent.
A full box can still miss
A box is only a count of packs. If a box holds 36 packs and the hit is 1 in 21 per pack, the all-miss chance is 17.3%, a bit more than one box in six. The expected number of hits is 1.71, and the average hides the spread: some boxes carry none and some carry three or more.
The formula assumes every pack is an independent draw at the stated odds. Some products use fixed slots, collation or guarantees, and then the assumption fails for that product. The TCGplayer data below shows one such structure. TCGplayer reports that 30th Celebration packs carry a guaranteed Pikachu Rare in the last slot of every pack, and that its other rarities sit in specific slots. For box-level guarantees, use the maker's statement for that product.
A measured rate is an estimate with an interval
TCGplayer's rates come from opened packs, so they carry sampling error. Its Authentication Center opened more than 3,000 booster packs of 30th Celebration and reported each rate with a 95 percent confidence interval. Its article reports that Special Illustration Rares came up in 4.88 percent of packs, plus or minus 0.74 points, which it rounds to 1 in 21. For one specific Special Illustration Rare card it gives 0.49 percent plus or minus 0.24 points, rounded to 1 in 205, and states that its per-card figures assume every card of a rarity is printed in equal numbers.
Run the interval through the formula for 10 packs and the answer is a range:
| Rate used | Chance of at least one Special Illustration Rare in 10 packs |
|---|---|
| Low end, 4.14 percent | 34.5% |
| Reported, 4.88 percent | 39.4% |
| High end, 5.62 percent | 43.9% |
The same 1 in 21 gives 38.6% when rounded first, close to the reported-rate row and below the high end. The interval, not the rounding, is the larger source of doubt. For a specific card at 0.49 percent, 36 packs give 16.2%, and the low and high ends of its interval, 0.25 and 0.73 percent, give 8.6% and 23.2%.
TCGplayer's Perfect Order article, based on more than 3,500 packs, states that it used a 95 percent interval with the normal approximation. It compares Ultra Rares in two sets, 8.06 percent plus or minus 0.75 in Phantasmal Flames against 8.54 percent plus or minus 0.92 in Perfect Order, and says the overlap points to ordinary sample variation rather than a change in production. A published rate is a measurement, so the second decimal place of any answer built on it carries no information.
What the Pokémon site states
The Pokémon TCG page on pokemon.com, read for this guide, lists no pull rates or odds of pulling any card. The rates quoted on this page come from TCGplayer's own counts. Where no rate is published for a product, the calculator has nothing to work from, and this page prints none.
Worked example with the calculator
Open the pack odds calculator, enter 21 as the odds, choose packs and enter 10. The result is 38.6% for at least one hit and 0.48 expected hits. The needed-packs rows show 15 packs for 50 percent, 48 for 90 percent and 62 for 95 percent. Change the unit to boxes, set packs per box to your product's count, and the box rows follow.
Sources
- TCGplayer, Pokémon TCG: 30th Celebration Pull Rates, published 24 September 2026, read 1 November 2026. Source of the packs opened, the rates, the intervals and the guaranteed last slot.
- TCGplayer, Pokémon TCG: Perfect Order Pull Rates, published 3 April 2026, read 1 November 2026. Source of the normal-approximation interval and the Ultra Rare comparison.
- The Pokémon Company, Pokémon TCG on pokemon.com, read 1 November 2026.
Short answers
- What are the odds of pulling a good Pokémon card?
- It depends on the set and the card, and the Pokémon site lists no pull rates. TCGplayer's count of more than 3,000 opened 30th Celebration packs put Special Illustration Rares at 4.88 percent of packs, plus or minus 0.74 points, which it rounds to 1 in 21.
- How many packs does it take to pull a card at 1 in 21?
- Ten packs give a 38.6 percent chance of at least one hit. Fifteen packs is the first count that reaches even odds, and 48 packs reach 90 percent.
- Can a full box still miss?
- Yes. At 1 in 21 per pack, a 36-pack box holds no hit 17.3 percent of the time, a bit more than one box in six. That assumes independent packs, and products with guaranteed slots break the assumption.
- Are published pull rates exact?
- No. TCGplayer's rates come from opened packs and carry a 95 percent confidence interval, so the second decimal place of any answer built on them carries no information.