Card Probability Calculator
Compute the probability of drawing a given number of favorable cards from a deck when sampling without replacement.
Chance of drawing one or more favorable cards
4
favorableFavorable cards
7.7%
Other cards
92.3%
- 1
Total possible hands C(N, n)
C(52, 5) = 2,598,960 - 2
Hands with no favorable card C(N−K, n)
C(48, 5) = 1,712,304 - 3
P(none) = C(N−K, n) ÷ C(N, n)
1,712,304 ÷ 2,598,960 = 0.658842 - 4
P(at least one) = 1 − P(none), as %
(1 − 0.658842) × 100 = 34.116Complement rule: P(at least one) = 1 − P(none of the favorable cards are drawn).
How does this calculator work?
For a deck of N cards with K favorable ones, drawing n without replacement, the chance of exactly k favorable cards is C(K,k)·C(N−K,n−k)/C(N,n), and the chance of at least one is 1−C(N−K,n)/C(N,n). Enter your deck size, favorable count, draws, and desired number to get both odds instantly.
Formula
How this is calculated
You describe a deck of N cards containing K favorable (success) cards, then draw n cards without replacement and ask about getting exactly k of the favorable cards. This is the hypergeometric distribution. The probability of exactly k successes is C(K,k)·C(N−K,n−k)/C(N,n): choose which k favorable cards appear, choose the remaining n−k from the N−K non-favorable cards, and divide by the total ways to choose n cards from N.
The probability of at least one favorable card is found via the complement: P(at least one) = 1 − P(none) = 1 − C(N−K,n)/C(N,n), since "none" means all n draws come from the N−K non-favorable cards. The expected number of favorable cards drawn equals n·(K/N). Binomial coefficients are evaluated through a log-gamma (Lanczos) approximation so large decks like a 52-card deck stay numerically stable.
All inputs are rounded to whole cards. The calculator requires N ≥ 1 with 0 ≤ K ≤ N and 0 ≤ n ≤ N; impossible requests (for example k greater than K, or needing more non-favorable cards than exist) yield a probability of zero. Probabilities are clamped to the [0, 1] range to absorb tiny floating-point error.
Frequently asked questions
Each card drawn is kept out of the deck, so later draws have fewer cards. This is the standard model for dealing a hand and uses the hypergeometric distribution rather than the binomial.
Set N = 52, K = 4 (the aces), n = 5, and read the "P(at least one)" value, which is about 34.1%.
It is zero whenever the request is impossible — for instance asking for more successes than there are favorable cards, or needing more non-favorable cards than the deck contains.
Also known as
TG we-Calculate Editorial Team. (2026). Card Probability Calculator [Online calculator]. TG we-Calculate. https://we-calculate.com/calculator/card-probability-calculator
TG we-Calculate Editorial Team. "Card Probability Calculator." TG we-Calculate. 2026. https://we-calculate.com/calculator/card-probability-calculator.
TG we-Calculate Editorial Team, "Card Probability Calculator," TG we-Calculate, 2026. [Online]. Available: https://we-calculate.com/calculator/card-probability-calculator
@misc{wecalculate_card_probability_calculator, title = {Card Probability Calculator}, author = {{TG we-Calculate Editorial Team}}, howpublished = {\url{https://we-calculate.com/calculator/card-probability-calculator}}, year = {2026}, note = {TG we-Calculate} }
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