Card Hand Probability Calculator
Our odds & chance calculator computes card hand probability instantly. Get useful results with practical tips and recommendations.
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer
Card Hand Probability Calculator
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?
About 42.3% for at least one pair in a 5-card hand. 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.
What is the difference between odds and probability?
Probability is expressed as a number between 0 and 1 (or a percentage), representing the likelihood of an event. Odds compare favorable outcomes to unfavorable ones — odds of 3:1 means 3 wins for every 1 loss, which is a probability of 3/(3+1) = 75%. Casinos often express odds differently from true probability to build in their house edge.
What is the probability of rolling a specific number on a standard die?
A fair six-sided die has 1/6 ≈ 16.67% probability for each face. Rolling at least one specific number in two rolls = 1 − (5/6)² ≈ 30.6%. Rolling two specific numbers on two dice = 1/36 ≈ 2.78%. These calculations multiply individual probabilities for independent events.
What is a fair game in probability theory?
A fair game is one where the expected value for all players is zero — no participant has a mathematical advantage. In practice, most casino games are unfair (negative EV for players) due to the house edge. Flipping a coin for even money is a fair game; flipping for $0.90 per win and $1 per loss is unfair.
How do poker hand probabilities work?
In a standard 52-card deck, there are 2,598,960 possible 5-card hands. Royal flush: 4 (0.000154%); straight flush: 36 (0.00139%); four of a kind: 624 (0.024%); full house: 3,744 (0.144%); flush: 5,108 (0.197%); straight: 10,200 (0.392%); three of a kind: 54,912 (2.11%); two pair: 123,552 (4.75%); one pair: 1,098,240 (42.3%); high card: 1,302,540 (50.1%).
What is the birthday problem in probability?
The birthday problem asks: how many people are needed for a 50% chance two share a birthday? The answer is just 23 people — surprising because there are 365 days. The probability no two people share a birthday with n people = (365/365)(364/365)(363/365)...(365−n+1)/365. With 23 people this equals ≈50.7%, meaning a shared birthday is more likely than not.
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer · Editorial policy
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