About the Card Draw Probability Calculator
This card draw probability calculator tells you how likely you are to draw a particular card — or a group of cards — from a shuffled deck. Enter the deck size, how many copies of the card are in it, how many cards you draw, and how many copies you want, and it returns the probability of drawing at least, exactly or at most that many, plus the full distribution from zero copies upwards.
Deck builders use it to tune Magic: The Gathering, Pokémon TCG, Yu-Gi-Oh!, Hearthstone and Lorcana decks: how often a 4-of shows up in an opening hand, how many lands you need to hit your drops, or whether a combo piece arrives by turn three. It works just as well for poker, bridge, blackjack and classroom probability problems.
The maths assumes a fairly shuffled deck and drawing without replacement — cards you draw are not put back — which is exactly the hypergeometric distribution. Mulligans, tutors and scry effects change the odds and are not included.
With the default inputs, the probability is 39.95%. Change any value above to recalculate instantly.
How to use the card draw probability calculator
- 1Enter the number of cards in your deck.
- 2Enter how many copies of the card (or how many cards that do the same job) it contains.
- 3Enter how many cards you will have seen — 7 for an opening hand, plus one per draw.
- 4Choose how many copies you want and whether you need at least, exactly or at most that many.
- 5Read the probability and use the table to see how the odds improve with each extra draw.
Formula and method
Drawing cards without putting them back follows the hypergeometric distribution. C(a, b) is the number of ways to choose b items from a. The formula counts the hands that contain exactly k of your K copies and n − k of the other N − K cards, then divides by every possible n-card hand from the deck.
“At least” adds up the exact probabilities for k, k + 1 … up to the maximum possible, which is easiest computed as one minus the chance of fewer than k. “At most” sums from zero to k. The average number of copies you draw is n × K ÷ N. Large combinations are handled with logarithms so decks of up to 1,000 cards stay accurate.
- N
- Total cards in the deck
- K
- Copies of the card (successes) in the deck
- n
- Cards drawn
- k
- Copies you want to draw
Worked examples
MTG: a 4-of in a 7-card opening hand
With 4 copies in 60 cards, the chance of no copies in 7 cards is C(56, 7) ÷ C(60, 7) = 60.05%, so the chance of at least one is 39.95%. On average you see 7 × 4 ÷ 60 = 0.47 copies.
24 lands: exactly 3 lands in the opening hand
A 60-card deck with 24 lands gives C(24, 3) × C(36, 4) ÷ C(60, 7) = 30.87% for exactly three lands in seven cards, and a 58.8% chance of three or more.
Poker: at least one ace in 5 cards
The chance of no aces in five cards is C(48, 5) ÷ C(52, 5) = 65.88%, so at least one ace turns up 34.12% of the time — about 1 hand in 2.93.
Frequently asked questions
What is the hypergeometric distribution?+
It gives the probability of a number of successes when you draw from a finite group without replacement — like drawing cards from a deck. Each card drawn changes the make-up of what is left, which is why it differs from simple coin-flip (binomial) odds.
What are the odds of drawing a 4-of in your opening hand in MTG?+
In a 60-card deck with 4 copies, the chance of at least one copy in your 7-card opening hand is about 39.9%. By your first draw on the play (8 cards) it rises to about 44.5%, and with 8 copies of similar cards it is about 65%.
How many lands should I run to hit my land drops?+
Most 60-card MTG decks run 22–26 lands. With 24 lands the chance of at least 2 lands in your opening hand is about 86%, and of at least 3 about 59%. Use the calculator with your land count and the number of cards seen by a given turn.
Does it matter if I am on the play or on the draw?+
Yes. On the draw you see one more card by each turn, so add one to the cards drawn. For example, by turn 3 you will have seen 9 cards on the play but 10 on the draw.
Does shuffling or drawing order change the probability?+
No. For a properly shuffled deck, the chance of a card being in any set of n positions is the same, so it does not matter whether you draw from the top, bottom or middle, or how the cards were dealt.
Can I calculate the odds of drawing two different cards?+
This calculator handles one group of cards at a time. For two different cards, multiply carefully or treat them as one group if either will do. If you need both, use a multivariate hypergeometric calculation.