🃏Card Probability Calculator (52-card deck)
e.g. 5 for poker
e.g. 4 aces
Card probability uses the Hypergeometric Distribution to calculate the probability of drawing specific cards from a standard 52-card deck without replacement. Unlike dice (with replacement), each card drawn changes the composition of the deck, making the probabilities hypergeometric rather than binomial.
- 1Hypergeometric: P(X=k) = C(K,k) × C(N−K, n−k) / C(N,n)
- 2N = deck size (52), K = cards of target type, n = hand size, k = desired count
- 3C(n,k) = n! / (k! × (n−k)!) — combinations formula
- 4P(at least k) = sum of P(X=k) + P(X=k+1) + ... + P(X=K)
5-card hand, probability of exactly 1 ace (K=4, n=5, k=1)=≈ 34.12%C(4,1)×C(48,4)/C(52,5)
5-card hand, at least 1 ace=≈ 34.12% + 21.37% + ... ≈ 34.12% (exactly 1) + others
7-card hand (Texas Hold'em + 2 in pocket), 2 aces=≈ 6.79% of starting hands are pocket aces
| Hand | Combinations | Probability |
|---|---|---|
| Royal flush | 4 | 0.000154% |
| Straight flush | 36 | 0.00139% |
| Four of a kind | 624 | 0.0240% |
| 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% |
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