Card Hand Probability Calculator
Our odds & chance calculator computes card hand probability instantly. Get useful results with practical tips and recommendations.
Card Hand Probability Calculator
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Formula: P = C(K,k)×C(N-K,n-k) / C(N,n) — Hypergeometric distribution
Worked example — 34.1% chance
Formula
P = C(K,k)×C(N-K,n-k) / C(N,n) — Hypergeometric distribution
The hypergeometric distribution gives the exact probability of drawing k target cards in n draws from a deck of N with K targets, without replacement.
Worked Examples
Example 1: Drawing an ace
Problem:5 cards from 52, need 1 of 4 aces
Solution:P(≥1 ace) = 1 - C(48,5)/C(52,5) = 34.1%
Result:34.1% chance
Frequently Asked Questions
What is the chance of getting a pair in poker?
Exactly one pair and nothing better comes up in about 42.3% of 5-card hands; counting every hand that contains a pair or better raises that to about 49.1%. Full house: 0.14%. Royal flush: 0.000154%.
What is the hypergeometric distribution and why does it apply to card games?
The hypergeometric distribution calculates the probability of drawing exactly k successes from a finite population without replacement. It applies to card games because each card drawn changes the composition of the remaining deck — unlike flipping coins, drawing cards is not independent. The formula C(K,k) × C(N-K, n-k) divided by C(N,n) gives the exact probability, where N is the deck size, K is the number of target cards, n is the hand size, and k is the desired number of target cards in hand.
How is card probability used in Magic: The Gathering deck building?
In collectible card games like Magic: The Gathering, players use hypergeometric probability to calculate the likelihood of drawing key cards in their opening hand (7 cards from 60) or within the first few turns. For example, to reliably draw at least one copy of a 4-of card in your opening hand, the probability is about 39.9%. Players use this math to decide how many copies of each card to include. Running more copies of a card increases consistency but reduces deck diversity. Tools like Card Hand Probability Calculator help optimize the number of lands, win conditions, and synergy pieces.
How does the number of decks in blackjack affect card probabilities?
In blackjack, using more decks reduces the impact of each card removed from the deck, making card counting less effective. With a single deck of 52 cards, removing one ace changes the remaining ace probability from 7.69% to 5.88% — a 24% relative change. With 8 decks (416 cards), removing one ace changes the probability from 7.69% to 7.47% — only a 2.9% relative change. This is why casinos use 6-8 deck shoes in blackjack: it flattens probability swings and makes advantage play much harder for card counters.
How do you calculate the chance of drawing at least one copy of a card?
Use the complement rule rather than summing every case. The number of hands that avoid all K target cards is C(N-K, n), so P(at least one) = 1 - C(N-K, n) / C(N, n). Drawing a 7-card opening hand from a 60-card deck holding 4 copies gives 1 - C(56,7)/C(60,7) = 39.9%. The complement takes one term instead of four, and it returns exactly the same answer as adding the exactly-one, exactly-two, exactly-three and exactly-four probabilities together.
What is the difference between the hypergeometric and binomial models for card draws?
The binomial distribution assumes a fixed success chance on every trial, which is only true if each card is shuffled back in. The hypergeometric distribution accounts for depletion: once you draw an ace, the next-ace chance falls from 4/52 to 3/51. Because the deck cannot keep supplying successes, hypergeometric variance is smaller, and binomial overstates the odds of several successes landing in the same hand. The two converge only when the sample n is a tiny fraction of the deck N.
How much do extra draws improve the chance of finding a key card?
What matters is the number of cards you actually see, so count draw spells and scry effects into n. In a 60-card deck with 4 copies, an opening hand of 7 finds at least one 39.9% of the time; by the tenth card seen it is 1 - C(56,10)/C(60,10) = 52.8%. Each additional card adds slightly less than the one before, because the misses you have already drawn are the cards being removed. Digging four cards deep early is worth roughly a dozen percentage points.
Why does deck size matter more than the raw number of copies?
The odds are driven by density K/N, not by K alone. Three copies in a 40-card limited deck beat four copies in a 60-card constructed deck for a 7-card opening hand: 1 - C(37,7)/C(40,7) = 44.8% against 39.9%. Every card added to the library dilutes every other card in it. That is why competitive players stay at the legal minimum deck size unless a specific card rewards a larger library, and why a 100-card singleton format is so much less consistent.
Why do poker players use the rule of four and two instead of this formula?
It is a mental shortcut built on outs, the unseen cards that complete a draw. On the flop, multiply outs by four for the approximate chance of hitting by the river; on the turn, multiply by two for the river alone. With nine outs for a flush the shortcut says 36%, while the exact hypergeometric figure is 1 - C(38,2)/C(47,2) = 35.0% from the 47 unseen cards. The approximation drifts high with many outs because it double counts hands that improve on both streets.
When should you read the exactly-k figure rather than the at-least-k figure?
Read exactly k when surplus copies are dead or actively harmful: a hand that wants precisely two lands, a combo a duplicate would spoil, or a card limited to one copy in play. Read at least k for keep-or-mulligan decisions, where any extra copy is simply fine. The two diverge by the whole upper tail. Dealt 5 cards from 52, exactly one ace is 29.9% while at least one ace is 34.1%, and the 4.2-point gap is entirely made up of hands holding two or more aces.
References
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer · Editorial policy
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