Card Probability Calculator
Work out the odds of drawing a single card, getting a certain mix in a hand, drawing cards in a set order, or hitting your key cards in a trading card game deck by a given turn. Every answer comes with the exact fraction and the working.
What do you want to work out?
For red cards and kings, that's the two red kings.
The whole hand is dealt before you look, so the order cards arrive in doesn't matter.
Uses a standard 52-card deck. Overlaps are handled for you, so "heart, then ace" correctly allows for the ace of hearts.
Groups can't share cards. Set copies to 0 to leave a group out. Works for Magic: The Gathering, Pokémon, Yu-Gi-Oh!, Lorcana and any game where you draw from a shuffled deck.
Results update as you type. Press Calculate or Enter to save one to your recent list.
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Why cards need different maths from dice and coins
A die has no memory. Roll a six and the next roll still has the same six faces to choose from. A deck of cards does remember. Once you take a card out and keep it, the deck is one card smaller and the mix of what's left has changed. If you pulled a heart, there are now fewer hearts waiting.
That's the whole reason card questions usually call for the hypergeometric distribution instead of the binomial one used for dice and coins. The two formulas look alike on paper, and for some questions they give close answers. For others they don't, and the gap can be bigger than most people expect.
The formulas behind each mode
One card
P(match) = matching cards / cards in the deck
There are 13 hearts in a 52-card deck, so the chance of drawing one is 13/52, which is 1/4 or 25%.
This or that
P(A or B) = (A + B − both) / N
Add the two groups, then take away the cards you counted twice. Red cards (26) or kings (4) is 26 + 4 − 2 = 28 cards, because the king of hearts and king of diamonds are in both groups. That gives 28/52, about 53.8%.
A hand (hypergeometric)
P(X = k) = C(K, k) × C(N − K, n − k) / C(N, n)
N is the deck size, K is how many matching cards are in it, n is the hand size and k is how many matches you want. C(K, k) counts the ways to pick your matches, C(N − K, n − k) counts the ways to fill the rest of the hand, and C(N, n) counts every possible hand.
One at a time, without putting cards back
P(all match) = K/N × (K − 1)/(N − 1) × (K − 2)/(N − 2) × ...
Each draw comes from a deck that's one card smaller. If the last card matched, there's also one fewer match left, so the top and bottom of each fraction both drop by one.
Deck builder (multivariate hypergeometric)
P = Σ C(c₁, x₁) × C(c₂, x₂) × C(rest, s − x₁ − x₂) / C(N, s)
The same idea as a hand, but with several groups at once. You add up every combination where each group meets its minimum. Here s is the number of cards you've seen by that turn: your opening hand plus one draw per turn, minus the turn 1 draw if you go first.
Worked examples
2 hearts in a 5-card hand
N = 52, K = 13, n = 5, k = 2. There are C(13, 2) = 78 ways to choose 2 hearts, and C(39, 3) = 9,139 ways to choose the other 3 cards from the 39 non-hearts. There are C(52, 5) = 2,598,960 possible 5-card hands in total.
So the answer is (78 × 9,139) / 2,598,960 = 712,842 / 2,598,960. That reduces to exactly 9,139/33,320, or about 27.43%.
3 hearts in a row
The first card is a heart 13 times in 52. If it is, 12 hearts remain among 51 cards. If the second is also a heart, 11 remain among 50. Multiply: 13/52 × 12/51 × 11/50 = 1,716/132,600, about 1.29%.
When the binomial shortcut goes wrong
For 2 hearts in a 5-card hand, the binomial formula gives 26.37% against the correct 27.43%. That's about one point off, which is easy to shrug at. Now try exactly 3 aces in a 13-card bridge hand. The correct hypergeometric answer is 4.12%. The binomial formula says 5.85%, which overstates the chance by more than 40%. The gap grows when you draw a large share of the deck, or when the cards you want are scarce, because each card you take changes the odds for the next one more sharply.
Lands in an opening hand
A 60-card Magic deck with 24 lands has a 58.79% chance of at least 3 lands in the opening 7. If you're allowed one mulligan that shuffles everything back and deals a fresh 7, the chance of seeing a hand like that at least once rises to 83.02%. A 40-card limited deck with 17 lands has an 89.48% chance of at least 2 lands in the opening 7.
Common card odds
All figures assume one standard 52-card deck, freshly shuffled.
5-card poker hands
| Hand | Ways to make it | Probability | About 1 in |
|---|---|---|---|
| Royal flush | 4 | 0.000154% | 649,740 |
| Straight flush (not royal) | 36 | 0.00139% | 72,193 |
| Four of a kind | 624 | 0.0240% | 4,165 |
| Full house | 3,744 | 0.144% | 694 |
| Flush (not a straight flush) | 5,108 | 0.197% | 509 |
| Straight (not a straight flush) | 10,200 | 0.392% | 255 |
| Three of a kind | 54,912 | 2.11% | 47 |
| Two pair | 123,552 | 4.75% | 21 |
| One pair | 1,098,240 | 42.3% | 2.37 |
| No pair | 1,302,540 | 50.1% | 2.00 |
Other situations
| Situation | Exact | Probability | About 1 in |
|---|---|---|---|
| Dealt pocket aces in Texas Hold'em | 1/221 | 0.452% | 221 |
| Dealt any pocket pair | 1/17 | 5.88% | 17 |
| Blackjack from the first two cards (single deck) | 32/663 | 4.83% | 20.7 |
| Completing a flush draw from the flop to the river | 378/1,081 | 34.97% | 2.86 |
| At least one ace in 5 cards | 18,472/54,145 | 34.12% | 2.93 |
| Exactly 3 aces in a 13-card bridge hand | 858/20,825 | 4.12% | 24.3 |
Mistakes that give the wrong answer
Using the binomial formula for a hand. The binomial formula assumes every draw has the same chance, which is only true if each card goes back and the deck is reshuffled. The bridge example above shows how far off that can be.
Mixing up a hand and a sequence. "2 hearts somewhere in my 5 cards" is a hand question. "A heart first, then a spade" is a sequence question. They start from the same deck but count different things, so they give different answers.
Adding overlapping groups. Every card has a suit and a rank at the same time. For "red card or king", simply adding 26 and 4 counts the two red kings twice. Use the This or that mode, which subtracts the overlap for you.
Forgetting the turn 1 draw rule in card games. In many games the player going first skips their first draw. By turn 3 you've seen 9 cards going first but 10 going second, and that single card can move the odds by several points.
How we calculate
Hand, sequence and deck builder results are worked out with exact whole-number arithmetic (JavaScript BigInt), so the fraction you see is exact, not an approximation. The percentage is rounded only at the last step, to three decimal places, or three significant figures for very small chances.
The simulation button deals 10,000 random hands or draws in your browser using a Fisher-Yates shuffle and counts how often your target happens. It will land close to the exact answer but not on it, and the result tells you the range to expect. That's normal random variation, not an error.
Every number in the worked examples and reference tables on this page was checked against an independent calculation in Python. If you spot a mistake, let us know and we'll correct it and note the change here.
References
- Sheldon Ross, A First Course in Probability, 10th edition, Pearson, 2019. Chapter 4 covers the hypergeometric distribution.
- William Feller, An Introduction to Probability Theory and Its Applications, Volume 1, 3rd edition, Wiley, 1968. Chapter 2 covers sampling without replacement and card hands.
- Charles Grinstead and J. Laurie Snell, Introduction to Probability, 2nd edition, American Mathematical Society, 1997. Freely available online.
Related calculators
- Binomial Probability Calculator, for draws where each card goes back in the deck
- Combinations and Permutations Calculator, which explains the C(n, r) term used throughout this page
- Probability Calculator, the full set, including dependent events and conditional probability
Questions people ask
Why can't I use the binomial formula for card hands?
The binomial formula assumes every draw has the same chance of success, which only holds if each card is put back and the deck reshuffled. Dealing a hand removes cards for good, so the odds change with every card. The hypergeometric distribution accounts for that.
How different are the binomial and hypergeometric answers?
It depends on how much of the deck you draw and how rare your target is. For 2 hearts in a 5-card hand the binomial answer is 26.37% and the correct answer is 27.43%. For exactly 3 aces in a 13-card bridge hand the binomial answer is 5.85% and the correct answer is 4.12%.
What's the difference between a hand and drawing one at a time?
A hand question only cares about which cards you end up holding, so it counts combinations. A one-at-a-time question cares about the order, such as a heart first and then a spade, so it multiplies a chain of changing fractions.
Does shuffling between hands reset the odds?
Yes. A full, freshly shuffled deck starts every calculation from scratch. The shrinking-deck maths only applies within a single deal.
How do I work out opening hand odds for Magic, Pokémon or Yu-Gi-Oh?
Use the Deck builder mode. Enter your deck size, opening hand size, how many copies of each card group you run and how many you need, then pick a turn and whether you go first or second. A 60-card deck with 24 lands has a 58.79% chance of at least 3 lands in the opening 7.
Why doesn't the simulation match the exact answer perfectly?
A simulation deals random hands, so its result wobbles a little each time you run it. With 10,000 trials it usually lands within about one percentage point of the exact answer. The exact answer is the one to rely on; the simulation is there so you can see it hold up in practice.
What are the odds of being dealt pocket aces?
There are 6 ways to be dealt two aces out of 1,326 possible two-card starting hands, so the chance is exactly 1 in 221, about 0.452%.