🃏 Card Deck Odds Calculator

Enter your deck size, number of target cards in the deck, and how many cards you draw to calculate the probability of drawing an exact number of target cards.

Deck Size

Target Cards in Deck

Cards Drawn

Desired Number of Target Cards

The hypergeometric formula behind the two numbers

Drawing is without replacement, so each card you take changes the deck for the next one. That is the hypergeometric distribution, with N the deck size, K the target cards in it, n the cards you draw and k the hits you want:

P(exactly k) = C(K,k) × C(N-K, n-k) / C(N,n)
P(at least k) = sum of P(exactly i) for i = k .. min(K,n)

The defaults ask for two aces in a five-card hand from a 52-card deck: C(4,2) × C(48,3) / C(52,5) = 6 × 17,296 / 2,598,960 = 3.993%. The at-least row adds the three-ace and four-ace cases, reaching 4.168%.

What each box means

Deck size is everything you could draw from, not just the part you have seen. Cards already in hand are no longer in the deck, so subtract them from both the deck size and the target count.
Target cards in deck counts every card that would satisfy you. Nine outs is nine, whether that is one card repeated or nine different cards doing the same job.
Desired number is a floor for the second row and an exact match for the first. Ask for more hits than you draw and the exact probability is zero, which is correct, not an error.

The at-least row is usually the one you want. "Do I see a land" is at least 1; "do I draw exactly one" is a different, smaller question.

Frequently asked questions

What are the odds of drawing an ace in a five-card hand?

34.116% for at least one ace, from deck size 52, 4 targets, 5 drawn, wanting 1. Exactly one ace is 29.947% - the gap is the hands holding two or more.

What are the odds of opening with a four-of in a 60-card deck?

39.950%. Enter 60, 4, 7 and 1: a seven-card opening hand contains at least one copy about two games in five, so a card you need every game usually wants more than four slots or a way to search for it.

Can I use this for dice rolls?

No. Dice and coin flips are drawn with replacement, so the odds never move between rolls. This tool assumes a finite deck that shrinks as you draw; use the dice probability calculator for rolls.