Hong Life CalculatorsUnderstanding the card draw probability calculator — concept and principle
In trading card games such as Magic: The Gathering, Pokémon TCG or Yu-Gi-Oh!, players want to know how likely they are to see a key card in the opening hand or by a certain turn. This calculator answers that with the hypergeometric distribution: given the deck size, how many copies of a card you run and how many cards you draw, it gives the chance of drawing at least a chosen number of copies.
How it is calculated
P(X ≥ k) = Σ from i = k to min(K, n) of C(K, i) × C(N − K, n − i) ÷ C(N, n)
- N: deck size. K: copies of the card. n: cards drawn. k: the minimum number of copies you want.
- C(a, b): the number of ways to choose b items from a, "a choose b".
- Each term counts hands with exactly i copies; the sum adds every case from k up to the most copies possible.
The principle
Every set of n cards from a shuffled deck is equally likely, so probability is a matter of counting. There are C(N, n) possible hands. A hand with exactly i copies must take i of the K copies and the remaining n − i cards from the N − K other cards, which can be done in C(K, i) × C(N − K, n − i) ways. Because cards are drawn without replacement, each draw changes what is left, which is why this is hypergeometric and not binomial. For "at least 1", the shortcut 1 − C(N − K, n) ÷ C(N, n) gives the same answer.
Worked example
- Inputs: deck 60, copies 4, cards drawn 7, at least 1.
- Chance of zero copies: C(56, 7) ÷ C(60, 7) = 0.6005.
- At least one: 1 − 0.6005 = 39.95%.
- Check by terms: exactly 1 = 33.63%, exactly 2 = 5.93%, exactly 3 = 0.38%, exactly 4 = 0.007%; together 39.95%.
| Deck size | Probability |
|---|---|
| 15 | 94.87% |
| 30 | 67.69% |
| 45 | 50.46% |
| 60 | 39.95% |
| 75 | 33% |
| 90 | 28.08% |
| 120 | 21.62% |
How to read the result
About 40% means a 4-of in a 60-card deck shows up in the opening 7 in two games out of five. Drawing one extra card (8 cards, as when going second in Magic) lifts this to 44.5%; by 10 cards it is 52.8%. A single copy appears in the opening 7 only 11.7% of the time, while 8 copies (two cards that do the same job) reach 65.4%. In a 40-card deck with 3 copies and a 5-card hand the chance is 33.8%. Use these figures to decide how many copies or redundant cards a deck needs.
Common mistakes
- Using the binomial formula, which assumes replacement and overstates the odds slightly.
- Confusing "exactly k" with "at least k".
- Forgetting cards already removed from the deck, such as prize cards in Pokémon or cards milled or exiled.
- Ignoring mulligans, which give extra chances to find a card.
Frequently asked questions
How many cards do I draw by turn 3?
In Magic, 7 for the opening hand plus one per turn: 9 on the play by turn 3, 10 on the draw.
What is the chance of drawing two specific different cards?
That needs a multivariate hypergeometric calculation; it is lower than either single chance.
Why not just multiply 4/60 by 7?
That gives 46.7%, which counts hands with several copies more than once. The exact answer is 39.95%.
Sources: William Feller, An Introduction to Probability Theory and Its Applications, Vol. 1, Wiley (hypergeometric distribution)