Playing Card Probability Calculator
Exact probability of drawing a specific number of target cards from a deck using the hypergeometric distribution. Covers poker, blackjack, and any card game.
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How card drawing probability works
The hypergeometric distribution gives the exact probability of drawing a specified number of "target" cards when cards are dealt from a deck without replacement. Because each card removed changes the composition of what remains, the draws are not independent — which distinguishes card drawing from a coin flip or a die roll, where each trial has a fixed probability. This calculator applies the distribution to any combination of deck size, number of target cards, hand size, and desired hit count.
The hypergeometric formula
For a deck of cards containing target cards, drawing cards without replacement, the probability of getting exactly hits is:
P(X=k)=(nN)(kK)(n−kN−K)Each term has a direct combinatorial meaning:
- — ways to choose exactly targets from the available
- — ways to fill the remaining slots with non-target cards
- — total ways to deal any cards from
The formula counts favorable hands and divides by all possible hands.
Worked example: two aces in a 5-card poker hand
Standard poker deal: $N = 52$, $K = 4$ (aces), $n = 5$, $k = 2$.
P(X=2)=(552)(24)(348)=25989606×17296≈0.03993Roughly 4.0% of 5-card hands contain exactly two aces. For at least one ace the probability is much higher — about 34.1%.
The expected number of aces per hand is:
E[X]=Nn⋅K=525×4=135≈0.385A hand averages just under half an ace, even though any single hand of course contains a whole number of aces — the expectation describes the long-run average across many deals.
At least k versus exactly k
The P(at least k) result — — sums the PMF from up to :
P(X≥k)=i=k∑min(n,K)P(X=i)The "at least" form answers a threshold question — the probability of drawing two or more hearts, say — whereas the PMF answers the narrower "exactly two." The cumulative result is always at least as large as the exact one, and the distribution chart shows how the probability mass is spread across all possible hit counts.
Common card-drawing scenarios
| Deck | Target | Hand | Desired | P(exactly) | P(at least) |
|---|---|---|---|---|---|
| 52 | 4 (aces) | 5 | 1 | 29.9% | 34.1% |
| 52 | 4 (aces) | 5 | 2 | 4.0% | 4.2% |
| 52 | 13 (hearts) | 5 | 3 | 8.2% | 9.3% |
| 52 | 12 (face cards) | 5 | 2 | 25.1% | 32.5% |
| 52 | 4 (aces) | 2 | 1 | 14.5% | 14.9% |
| 312 (6-deck) | 24 (aces) | 2 | 1 | 14.2% | 14.8% |
The six-deck blackjack shoe row shows that when is held constant (24/312 = 4/52) the probability barely shifts — the large denominator dominates.
Hypergeometric vs. binomial
The binomial distribution applies when each trial is independent with a fixed probability . Card draws are not independent — removing a card changes for the next draw. For a 52-card deck the difference is small but present:
- Binomial approximation for 1 ace in 5 cards:
- Exact hypergeometric:
The discrepancy grows when the hand is a large fraction of the deck. For a 10-card hand from a 20-card deck the binomial approximation breaks down noticeably; the hypergeometric formula remains exact.
Using this calculator
- Deck Size — total cards before any draws. Standard deck: 52. Six-deck shoe: 312. Remove already-dealt cards to model mid-game situations.
- Target Cards in Deck — how many cards count as "hits." Aces: 4. Spades: 13. Red kings: 2.
- Hand Size — cards drawn in one deal.
- Desired Hits — the exact count being queried. The P(at least k) result covers a minimum threshold rather than an exact count.
The distribution chart visualises the full PMF — how probability mass is spread from zero hits to the maximum possible. The highlighted bar corresponds to the Desired Hits input.
Frequently Asked Questions (FAQ)
What is the probability of getting two aces in a 5-card hand?
With a standard 52-card deck, 4 aces, and a 5-card hand: P(X = 2) = C(4,2) × C(48,3) / C(52,5) = 6 × 17,296 / 2,598,960 ≈ 0.03993, or about 4.0%. The chance of at least one ace is higher at about 34.1%. Enter Deck = 52, Target = 4, Hand = 5, Desired = 2 to confirm.
Why is card drawing a hypergeometric distribution?
The hypergeometric distribution models sampling without replacement from a finite population split into two groups.
Drawing cards fits perfectly: the deck is the population, the target cards are one group, and the rest are the other. Each card removed changes the composition for the next draw — the defining feature of sampling without replacement. If you returned each card before the next draw, the simpler binomial distribution would apply instead.
Does this calculator assume drawing with or without replacement?
Without replacement — this is the standard rule for all card games. Once a card is drawn it is not returned to the deck, so the probabilities shift with each draw. The hypergeometric formula accounts for this exactly. If you need with-replacement probabilities (each card returned before the next draw), the binomial formula applies and you can use the Binomial Probability Calculator.
How do I compute the probability of a full poker hand like a flush?
For multi-condition hands you need to count favorable 5-card combinations directly rather than using a single hypergeometric calculation.
For example, a flush (5 cards of the same suit): C(13,5) = 1,287 ways per suit × 4 suits = 5,148 flush hands out of C(52,5) = 2,598,960 total hands ≈ 0.197%. This calculator handles single-condition draws (exactly or at least k cards of one type); for full poker hand frequencies refer to a combinatorics reference or poker odds table.
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