Card Hand Probability Calculator
Our odds & chance calculator computes card hand probability instantly. Get useful results with practical tips and recommendations.
Reviewed by Daniel Agrici, Founder & Lead Developer
Formula
P = C(K,k)×C(N-K,n-k) / C(N,n) — Hypergeometric distribution
The hypergeometric distribution gives the exact probability of drawing k target cards in n draws from a deck of N with K targets, without replacement.
Worked Examples
Example 1: Drawing an ace
Problem:5 cards from 52, need 1 of 4 aces
Solution:P(≥1 ace) = 1 - C(48,5)/C(52,5) = 34.1%
Result:34.1% chance
Frequently Asked Questions
What is the difference between odds and probability?
Probability is expressed as a number between 0 and 1 (or a percentage), representing the likelihood of an event. Odds compare favorable outcomes to unfavorable ones — odds of 3:1 means 3 wins for every 1 loss, which is a probability of 3/(3+1) = 75%. Casinos often express odds differently from true probability to build in their house edge.
What is the probability of rolling a specific number on a standard die?
A fair six-sided die has 1/6 ≈ 16.67% probability for each face. Rolling at least one specific number in two rolls = 1 − (5/6)² ≈ 30.6%. Rolling two specific numbers on two dice = 1/36 ≈ 2.78%. These calculations multiply individual probabilities for independent events.
What is a fair game in probability theory?
A fair game is one where the expected value for all players is zero — no participant has a mathematical advantage. In practice, most casino games are unfair (negative EV for players) due to the house edge. Flipping a coin for even money is a fair game; flipping for $0.90 per win and $1 per loss is unfair.
How do poker hand probabilities work?
In a standard 52-card deck, there are 2,598,960 possible 5-card hands. Royal flush: 4 (0.000154%); straight flush: 36 (0.00139%); four of a kind: 624 (0.024%); 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%).
Reviewed by Daniel Agrici, Founder & Lead Developer · Editorial policy