Card Draw Probability Calculator

How it is calculated

You enter the deck size N, the number of target cards still in it K, the number of cards drawn n and the minimum number of targets you want k. Cards are drawn without replacement, so the chance of exactly i targets is C(K,i) × C(N−K, n−i) ÷ C(N,n). The tool reports two numbers: the probability of exactly k targets, and the probability of at least k, which is the sum of the terms for i = k up to the largest possible count. Decimals are cut down to whole numbers. The inputs must satisfy K ≤ N and n ≤ N, otherwise there is no result. Every card is treated as equally likely to be drawn from a well-shuffled deck.

Common mistakes

• Reading the exactly figure when you mean at least. For wanting one copy, the two numbers are 33.63% and 39.95% in this case, and they answer different questions.
• Entering the total number of target cards you own instead of the number still left in the deck. After earlier draws, K and N must both be updated.
• Counting the cards drawn as the number of turns. If you draw 7 in the opening hand and 1 per turn, the number of draws on turn 3 is 7 + 2 = 9, not 3.
• Applying the result to a deck that is not random, for instance after searching it or with a known top card. The formula assumes a fair shuffle.
• Treating the odds as a promise. A 39.95% chance still fails in 6 out of 10 hands on average.