Last updated: 2026-05-19

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 $N$ cards containing $K$ target cards, drawing $n$ cards without replacement, the probability of getting exactly $k$ hits is:

Each term has a direct combinatorial meaning:

• $\binom{K}{k}$ — ways to choose exactly $k$ targets from the $K$ available
• $\binom{N-K}{n-k}$ — ways to fill the remaining $n-k$ slots with non-target cards
• $\binom{N}{n}$ — total ways to deal any $n$ cards from $N$

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), $n = 5$, $k = 2$.

Roughly 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:

A 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 — $P(X \geq k)$ — sums the PMF from $k$ up to $\min(n, K)$:

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

The six-deck blackjack shoe row shows that when $K/N$ 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 $p$. Card draws are not independent — removing a card changes $p$ for the next draw. For a 52-card deck the difference is small but present:

• Binomial approximation for 1 ace in 5 cards: $1 - (1 - 4/52)^5 \approx 33.0\%$
• Exact hypergeometric: $\approx 34.1\%$

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

1. Deck Size — total cards before any draws. Standard deck: 52. Six-deck shoe: 312. Remove already-dealt cards to model mid-game situations.
2. Target Cards in Deck — how many cards count as "hits." Aces: 4. Spades: 13. Red kings: 2.
3. Hand Size — cards drawn in one deal.
4. 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.