Card Probability Calculator
Calculate the probability of drawing specific cards from a deck. Uses hypergeometric and binomial distributions for exact probabilities with or without
Card Probability Calculator
Calculator
Adjust values & calculateEnter your values below. Every result is computed in your browser — no data is sent to any server.
Formula: P(X=k) = C(K,k) x C(N-K,n-k) / C(N,n)
Worked example — P(at least 1 ace) = 34.12%
Formula
P(X=k) = C(K,k) x C(N-K,n-k) / C(N,n)
The hypergeometric probability formula calculates the chance of drawing exactly k desired cards when n cards are drawn from a deck of N total cards containing K desired cards, without replacement. C(n,r) represents the combination function (n choose r).
Worked Examples
Example 1: Drawing at Least One Ace in 5 Cards
Problem:What is the probability of drawing at least one ace when dealt 5 cards from a standard 52-card deck without replacement?
Solution:Deck size N = 52, Desired cards K = 4 (aces), Draw count n = 5 P(no aces) = C(48,5) / C(52,5) C(48,5) = 1,712,304 C(52,5) = 2,598,960 P(no aces) = 1,712,304 / 2,598,960 = 0.6588 P(at least 1 ace) = 1 - 0.6588 = 0.3412
Result:P(at least 1 ace) = 34.12%
Example 2: Drawing Exactly 2 Hearts in 7 Cards
Problem:What is the probability of drawing exactly 2 hearts from a 52-card deck when drawing 7 cards without replacement?
Solution:N = 52, K = 13 (hearts), n = 7, k = 2 P(X=2) = C(13,2) x C(39,5) / C(52,7) C(13,2) = 78 C(39,5) = 575,757 C(52,7) = 133,784,560 P(X=2) = 78 x 575,757 / 133,784,560 P(X=2) = 44,909,046 / 133,784,560 = 0.3356
Result:P(exactly 2 hearts) = 33.56%
Frequently Asked Questions
How does the hypergeometric distribution apply to card drawing?
The hypergeometric distribution models the probability of drawing a specific number of success cards from a finite deck without replacement. This is the most accurate model for standard card games because once a card is drawn, it is not returned to the deck, changing the composition of the remaining cards. The formula is P(X=k) = C(K,k) x C(N-K,n-k) / C(N,n), where N is the total deck size, K is the number of desired cards in the deck, n is the number of draws, and k is the number of desired cards you want to draw. This differs from the binomial distribution which assumes each draw is independent, applicable only when cards are replaced after each draw. Understanding this distinction is essential for accurate probability calculations in poker, blackjack, and trading card games.
What is the probability of being dealt a specific poker hand?
Poker hand probabilities are calculated using combinations from a standard 52-card deck with 5 cards dealt. A royal flush has only 4 possible combinations out of 2,598,960 total five-card hands, giving a probability of 0.000154 percent or roughly 1 in 649,740. A straight flush excluding royal flush has 36 combinations for 0.00139 percent probability. Four of a kind has 624 combinations at 0.024 percent. A full house has 3,744 combinations at 0.144 percent. A flush has 5,108 combinations at 0.197 percent. A straight has 10,200 combinations at 0.392 percent. Three of a kind occurs at 2.11 percent, two pair at 4.75 percent, and one pair at 42.26 percent. No matching hand occurs about 50.12 percent of the time.
What is the difference between drawing with and without replacement?
Drawing without replacement means each card drawn is removed from the deck, reducing both the total cards remaining and potentially the number of desired cards remaining. This makes each subsequent draw dependent on all previous draws. For example, if you draw an ace from a 52-card deck, the probability of drawing another ace changes from 4/52 to 3/51. Drawing with replacement means the card is returned to the deck after each draw, and the deck is reshuffled, keeping probabilities constant across all draws. Without replacement uses the hypergeometric distribution while with replacement uses the binomial distribution. Most real card games use without replacement, but some probability exercises, shuffle machines, and theoretical scenarios use with replacement for simplified calculations.
How do I calculate the odds of drawing at least one desired card?
The easiest way to calculate the probability of drawing at least one desired card is to use the complement method. Instead of calculating P(at least 1) directly, which requires summing many individual probabilities, calculate P(none) and subtract from 1. For drawing without replacement, P(none) = C(N-K, n) / C(N, n), where N is deck size, K is desired cards, and n is draw count. Then P(at least 1) = 1 - P(none). For example, the probability of drawing at least one ace in a 5-card hand from a standard deck is 1 - C(48,5)/C(52,5) = 1 - 1,712,304/2,598,960 = 34.12 percent. This complement approach works for any distribution and is computationally much simpler than summing probabilities for exactly 1, exactly 2, exactly 3, and exactly 4 successes.
How are card probabilities used in competitive card games and gambling?
Professional card players and gambling analysts use probability calculations extensively to make optimal decisions. In poker, pot odds compare the probability of completing a drawing hand against the ratio of the current pot to the cost of a call. If your probability of winning exceeds the pot odds percentage, calling is mathematically profitable in the long run. In blackjack, card counting systems track the ratio of high to low cards remaining, adjusting bet sizes when the probability distribution favors the player. In collectible card games like Magic the Gathering, deck builders use hypergeometric probability to determine the optimal number of copies of each card type to ensure consistent draws. Expected value calculations combine probabilities with potential payoffs to guide strategic decisions across all competitive card formats.
What does the combination function C(n,r) actually count?
C(n,r), read as n choose r, counts how many distinct groups of r items can be selected from n when order does not matter. It equals n! divided by r! times (n-r)!. A five-card poker hand from a 52-card deck gives C(52,5) = 2,598,960 possible hands, because the same five cards dealt in a different sequence form the same hand. Combinations sit at the heart of the hypergeometric formula, which multiplies the ways of choosing your successes by the ways of choosing the remaining non-success cards and divides by the total ways of choosing the draw.
How do outs and the rule of four and two work in Texas hold'em?
An out is any unseen card that completes your hand. With a flush draw after the flop you hold four of a suit, leaving nine outs among the 47 unseen cards. The shortcut is to multiply outs by four for two cards to come and by two for one card. Nine outs gives roughly 36 percent by the shortcut, and the exact hypergeometric answer is 1 minus (38/47) times (37/46), which is about 35.0 percent. The approximation drifts high for large out counts, so recompute exactly when the pot is large.
How do you find the expected number of desired cards in a draw?
The mean of the hypergeometric distribution is simply n times K divided by N, where n is how many cards you draw, K is how many desired cards are in the deck and N is the deck size. Drawing seven cards from a 60-card deck containing 17 lands gives an expected 7 times 17 divided by 60, about 1.98 lands. The expectation needs no combinations at all, but it says nothing about spread, so you still need the full distribution to know how often you will be badly short.
How is hypergeometric probability used when building a deck?
Deck builders choose copy counts by asking how often a card appears in the opening hand or by a given turn. Increasing the copies of a card increases K in the formula, and drawing deeper increases n, so both levers raise the chance of seeing it. Because the curve of that probability flattens, the jump from one copy to two is much larger than the jump from three to four. Running the numbers for each key card lets you balance consistency against the risk of drawing redundant copies late in the game.
How should you update card probabilities after seeing exposed cards?
Every card you can see, whether in your hand, on the board or in a discard pile, must be removed from both the deck size N and, if it is a desired card, from the success count K. This is conditional probability applied to a shrinking deck. If two aces are already visible, the chance the next card is an ace uses 2 remaining aces over the reduced number of unseen cards, not 4 over 52. Failing to condition on visible information is the most common source of error in live probability estimates.
References
Reviewed for accuracy by Daniel Agrici, Founder & Lead Developer · Editorial policy
Related Calculators
🧮Card Hand Probability Calculator
Calculate card hand probability with inputs, formulas, and instant results.
🧮Bridge Hand Evaluator
Evaluate a bridge hand using high card points, distribution points, and trick-taking potential.
🧮Gacha Probability Calculator
Calculate gacha probability with inputs, formulas, and instant results.
🧮Match Win Probability Calculator
Calculate match win probability with inputs, formulas, and instant results.
🧮Dice Probability Calculator
Calculate the probability of rolling specific outcomes with any number and type of dice.
🧮Coin Toss Probability Calculator
Calculate the probability of getting a specific sequence of heads and tails in multiple flips.
🧮Gaming Pc Fps Calculator
Estimate frames per second based on GPU, CPU, resolution, and game optimization level.
🧮Dice Odds Calculator
Calculate dice odds with inputs, formulas, and instant results.