How to Calculate Card Probability

What is Card Probability?

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.

Formula

N

n! / (k! × (n−k)!) — combinations formula — n! / (k! × (n−k)!) — combinations formula

K

sum of P(X=k) + P(X=k+1) + — sum of P(X=k) + P(X=k+1) +

n

n! / (k! × (n−k)!) — combinations formula — n! / (k! × (n−k)!) — combinations formula

k

sum of P(X=k) + P(X=k+1) + — sum of P(X=k) + P(X=k+1) +

Step-by-Step Guide

1. 1Hypergeometric: P(X=k) = C(K,k) × C(N−K, n−k) / C(N,n)
2. 2N = deck size (52), K = cards of target type, n = hand size, k = desired count
3. 3C(n,k) = n! / (k! × (n−k)!) — combinations formula
4. 4P(at least k) = sum of P(X=k) + P(X=k+1) + ... + P(X=K)

Worked Examples

Frequently Asked Questions