Card draw probability calculator
Drawing without replacement follows the hypergeometric distribution, which is the right model for cards, and the wrong one for dice.
Chance of exactly that many
29.947 36%
- Chance of at least one
- 34.115 80 %
- Chance of none
- 65.884 20 %
- Expected number drawn
- 0.384 62
- Possible hands
- 2 598 960
- Hands with exactly that many
- 778 320
- One in how many hands
- 3.34
Show the working
- Cards in the deck = 52
- Cards of the type you want = 4
- Cards drawn = 5
- How many you want = 1
- Possible hands = comb(52, 5) = 2 598 960
- Hands with exactly that many = comb(4, 1) * comb(52 - 4, 5 - 1) = 778 320
- Chance of exactly that many = 2 598 960 > 0 ? 778 320 / 2 598 960 * 100 : 0 = 29.9474
- none_ways = comb(52 - 4, 5) = 1 712 300
- Chance of none = 2 598 960 > 0 ? 1 712 300 / 2 598 960 * 100 : 0 = 65.8842
- Chance of at least one = 100 - 65.8842 = 34.1158
- Expected number drawn = 52 > 0 ? 5 * 4 / 52 : 0 = 0.384 615
- One in how many hands = 29.9474 > 0 ? 100/29.9474 : 0 = 3.339 19
Values are shown to six significant figures here; the arithmetic itself runs on exact fractions and is only rounded where a number is printed.
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The formula
- P(X = k) = C(K,k) C(N-K, n-k) / C(N,n)
- with N the deck size, K the successes in the deck and n the draws
Drawing five cards from a standard deck, the chance of at least one ace is 34.1%.