Card Draw Probability Calculator
When you draw cards from a deck without shuffling them back, the probability of drawing a specific card follows the hypergeometric distribution. This calculator computes the probability of drawing exactly k copies of a card given your deck size, number of copies in the deck, hand size, and desired copies in hand. It also shows the probability of drawing at least 1 copy. Essential for card game design and deck optimization in games like Magic: The Gathering, Pokemon, and similar collectible card games.
Hypergeometric probability formula
P(X = k) = C(K, k) * C(N-K, n-k) / C(N, n)
P(X >= 1) = 1 - P(X = 0) = 1 - C(N-K, n) / C(N, n)
N = deck size, K = copies of card in deck, n = hand size, k = desired copies. C(a, b) = a! / (b! * (a-b)!) is the binomial coefficient (combinations).
Card probability in deck building
- With 4 copies in a 60-card deck, drawing at least 1 copy in a 7-card opening hand is approximately 39.9%.
- Running more copies dramatically increases consistency. Going from 2 to 4 copies roughly doubles your chance of seeing the card in the opening hand.
- Deck tutors and card search effects are valued precisely because they manipulate effective copy count and drawing probability.
- The hypergeometric distribution also applies to drafting, where you draw from a shared pool rather than a personal deck.
Card draw probability: frequently asked questions
What probability distribution applies to card draws?
Card drawing without replacement follows the hypergeometric distribution. Unlike the binomial distribution (which assumes replacement), the hypergeometric accounts for the fact that drawing one card changes the composition of the remaining deck.
What does the hypergeometric probability formula look like?
P(X = k) = C(K, k) * C(N-K, n-k) / C(N, n), where N is deck size, K is copies of the target card in the deck, n is hand size (cards drawn), and k is the desired number of copies in hand. C(a, b) denotes combinations.
How do I use this for Magic: The Gathering or similar games?
Enter deck size (typically 60 or 40), number of copies of the card in your deck (1 to 4), hand size (7 for opening hand, more for draws), and the number of copies you want. The result is the probability of exactly that many copies appearing.
What is the probability of drawing at least k copies?
The calculator shows P(X = k). For at least k, sum probabilities for k, k+1, k+2, and so on up to the minimum of K and n. The probability of drawing at least 1 is 1 minus the probability of drawing 0 copies.
Why is hypergeometric different from just dividing copies by deck size?
Dividing copies by deck size gives the probability on a single draw. For multiple draws without replacement, each subsequent draw is affected by earlier ones, making the combined probability non-linear. The hypergeometric formula handles this exactly.
Official sources
- NIST/SEMATECH e-Handbook of Statistical Methods, Hypergeometric Distribution: itl.nist.gov/div898/handbook/.
- National Council of Teachers of Mathematics: nctm.org.
Reviewed by the CalculatorHub team, edited by James Graham, 15 June 2026. See our methodology.