Card draw probability uses the hypergeometric distribution, which models drawing without replacement from a finite population. Unlike the binomial distribution (which assumes replacement), the hypergeometric distribution accounts for the changing composition of the deck as cards are drawn. This makes it the correct model for poker, blackjack, and any probability of drawing a card scenario. Our card probability calculator implements this distribution with full step-by-step work.
The Formula
Where N = deck size, K = total successes in deck, n = cards drawn, k = desired successes. The formula combines three combinatorial terms: ways to choose k successes from the K in the deck, ways to choose the remaining n−k cards from the N−K non-successes, divided by the total ways to choose n cards from the full deck.
Worked Example
What's the probability of drawing a card — specifically exactly 1 ace in a 5-card poker hand? N=52, K=4 (four aces), n=5 (hand size), k=1 (one ace). C(4,1) = 4, C(48,4) = 194,580, C(52,5) = 2,598,960. So P = 4 × 194,580 / 2,598,960 = 778,320 / 2,598,960 ≈ 0.2995 = 29.95%.
Deck Presets
Our calculator includes preset configurations for popular decks: Standard 52 (poker deck, 4 suits of 13), Pinochle (48) (two copies of 9-A in 4 suits), Euchre (24) (9-A in 4 suits), and Tarot (78) (56 minor arcana + 22 major arcana). The Custom preset lets you set any deck size and success count for non-standard games.
Draw Sequence Mode
What's the probability of drawing a specific sequence of cards without replacement? For example, drawing an Ace then a King from a standard deck: P = (4/52) × (4/51) ≈ 0.60%. Each subsequent draw changes the deck composition, so the probabilities are multiplied sequentially. This mode handles two-card sequences with any deck size and copy counts.
Poker and Blackjack Applications
Card probability is essential for serious poker players. In Texas Hold'em, if you have four cards to a flush after the flop, there are 9 remaining cards of that suit out of 47 unseen cards. The probability of completing your flush on the next card is 9/47 ≈ 19.1%. Over the turn and river combined, P(hit flush) = 1 − (38/47 × 37/46) ≈ 35%.