Card Draw Probability Calculator
Calculate the probability of drawing at least a certain number of desired cards when drawing from a deck (hypergeometric distribution).
Results will appear here.
Formula
This calculator uses the Hypergeometric Distribution:
P(X = k) = C(K, k) × C(N−K, n−k) / C(N, n)
- N = total cards in the deck
- K = total desired cards in the deck
- n = number of cards drawn
- k = desired number of successes in the draw
- C(a, b) = binomial coefficient = a! / (b! × (a−b)!)
Mean: μ = n × (K / N)
Variance: σ² = n × (K/N) × ((N−K)/N) × ((N−n)/(N−1))
Log-space computation is used to handle large factorials accurately.
Assumptions & References
- Drawing is done without replacement (hypergeometric model). For drawing with replacement, use the binomial distribution instead.
- All cards are equally likely to be drawn at any point.
- The deck is well-shuffled (uniform random distribution).
- N, K, n, k must all be non-negative integers with K ≤ N and n ≤ N.
- Floating-point precision: log-space arithmetic is used to avoid overflow for large decks.
- Reference: Hypergeometric distribution — Feller, W. (1968). An Introduction to Probability Theory and Its Applications.
- Common use cases: trading card games (MTG, Pokémon), poker hand analysis, quality control sampling.