Enter your deck size and how many copies you run. You'll get the exact chance of seeing at least one in your opening hand, the full spread from zero to four, your odds by any turn on the play or the draw, and what mulligans honestly buy.
Drawing without replacement, done exactly
Every card you see changes what is left in the deck, which is why deck math is the hypergeometric distribution rather than coin-flip multiplication. This calculator computes it exactly: the chance of at least one copy in your opener, the full exactly-0-through-4 spread, the odds by any turn on the play or the draw, and what chasing a card through London mulligans really buys. It works for any "group of cards" question, not just playsets: 17 lands in a 40-card limited deck is the same math wearing different clothes.
The number that quietly matters most is the expected copies decimal. A playset in a 60-card deck averages 0.47 copies per opening hand, and that expectation, not the at-least-one percentage, is the foundation manabase math is built on: it is how you go from "how often do I see a land" to "how many sources of white do I need," which is exactly where our upcoming land calculator picks up.
The formula
P(exactly k) = C(K, k) × C(N − K, n − k) ÷ C(N, n)
Expected copies = n × K ÷ N
N is the deck size, K the copies, n the cards you have seen, and C the binomial coefficient ("choose"). On the play you have seen 7 + (turn − 1) cards, on the draw 7 + turn, because the player on the play skips their first draw step.
Worked example
A playset: 4 copies in a 60-card deck, 7-card hand. At least one copy: 39.95%: almost exactly the coin flip players intuit, but reached honestly. The spread: 33.63% for exactly one, 5.93% for two, 0.38% for three, and one hand in about 14,000 holds all four. Expected copies: 0.47.
By turn 4 you have seen 10 cards on the play (52.77%) or 11 on the draw (56.55%): the draw's extra card is worth 3.78 points here. And if you would ship any hand without it, two London mulligans of 7 fresh cards each raise the chance of finding it to 78.35%, at the price of keeping just 5 cards.
Three readings players get wrong
The expected count is not a probability. Multiplying 4/60 by 7 gives 0.47, and it is tempting to read that as 47%; the true at-least-one chance is 39.95%, because the lucky multi-copy hands are hiding inside the average. The two numbers answer different questions, and the table above shows both. The play/draw gap is real but small for any single card: about 3 to 4 points by the mid-game; the draw's real gift is card economy, not finding one specific answer. And the London mulligan is a combo player's tool, not a free reroll: each throw genuinely shows 7 fresh cards (the math here assumes you ship every hand without your card), but paying a card per mulligan means the rest of your hand gets worse while your odds of the one card get better. That trade is worth it for a piece your deck cannot function without, and for very little else, which is exactly what the 78% figure in the example is quietly telling you.
Frequently asked questions
What are the odds of drawing a 4-of in my opening hand?
39.95% for at least one copy of a playset in a 60-card deck with a 7-card hand: 33.63% for exactly one, 5.93% for two, 0.38% for three, and about 1 hand in 14,000 holds all four. Close to a coin flip, but reached by exact hypergeometric math rather than intuition.
What is the hypergeometric distribution and why does deck math need it?
It is the probability of drawing without replacement: every card you see changes what remains, so coin-flip multiplication is wrong from the second card onward. The formula counts the hands containing exactly k copies against all possible hands, which is why this page's answers are exact rather than simulated.
How much better are my draws on the draw than on the play?
One extra card: by turn 4 you have seen 11 cards instead of 10, which lifts the chance of finding a specific playset from 52.77% to 56.55%. For any single card the gap stays around 3 to 4 points; the draw's real advantage is total card economy rather than finding one answer.
How does the London mulligan change my odds of finding a card?
Each mulligan draws 7 fresh cards from the full reshuffled deck, so chasing one card through two throws means three independent 7-card looks: 78.35% for a playset, up from 39.95%. The price is real: you keep one fewer card per mulligan, so the math favors doing this only for a piece your deck cannot function without.
What is expected copies and why should I care about a decimal?
It is the average number of copies per hand: cards seen times copies divided by deck size, so a playset averages 0.47 per 7-card hand. It is not a probability, but it is the foundation of manabase math, where questions like how many white sources you need are answered in expectations, not coin flips.
Can I use this for lands instead of a single card?
Yes; copies just means any group of cards. Enter 17 copies in a 40-card limited deck and the table shows an 89.48% chance of at least two lands in your opener and a 1.32% chance of a landless hand. The same trick answers questions about removal counts, creatures, or any category you can count.
Do these odds work for Commander?
Yes: set the deck to 99 and copies to 1 for any singleton, which gives a 7.07% chance of seeing a specific card in your opener. The math is why Commander leans so hard on tutors and card draw: a 1-of is simply rare in any 7 cards, no matter how important it is to your plan.
Why does multiplying copies by hand size give the wrong probability?
Because that product is the expected count, not a probability: 4/60 times 7 is 0.47, which reads temptingly like 47%, while the true at-least-one chance is 39.95%. The average includes the lucky hands holding two or three copies, and those hide the gap. The two numbers answer different questions, so the page shows both.