What is the hypergeometric distribution?

It is the probability model for drawing without replacement from a finite population. In Magic, your deck is the population, the copies of a card are the successes, and your draw is the sample. It is the correct model for card draws because once you draw a card it cannot be drawn again.

How is the probability calculated?

The exact probability of drawing k copies is C(K, k) times C(N minus K, n minus k) divided by C(N, n), where N is deck size, K is copies in deck, n is cards drawn, and C is the binomial coefficient. The tool sums these terms to produce at-least and at-most figures.

Why not just use a simple percentage?

Because draws are not independent. After you draw one copy, fewer copies and fewer total cards remain, so the odds of the next draw change. The hypergeometric distribution accounts for this exactly, while a naive percentage does not.

Can I use this for the opening hand?

Yes. Set cards drawn to 7 for a typical opening hand. To model later turns, increase the draw count by one per turn drawn. The calculation is identical for any sample size up to the deck size.

Is this accurate for large Commander decks?

Yes. The tool uses log-factorials internally, so it stays numerically stable and accurate for decks of 100 cards or more without overflow errors.

MTG Hypergeometric Draw Calculator

The hypergeometric distribution is the foundation of nearly every Magic deckbuilding probability question: how likely are you to draw a removal spell by turn three, to open with at least two lands, or to hit your combo piece in your first ten cards. This calculator gives you the exact answer for any deck size, copy count, and draw count.

How it works

You provide four numbers: deck size N, copies of the wanted card in the deck K, cards drawn n, and copies wanted k. The probability of drawing exactly k copies is:

P(X = k) = C(K, k) * C(N - K, n - k) / C(N, n)

From that single formula the tool builds three useful figures:

Exactly k — the formula above evaluated at your chosen k.
At least k — one minus the sum of the probabilities for 0 through k - 1 copies.
At most k — the sum of the probabilities for 0 through k copies.

The expandable distribution table lists the probability of every possible outcome, from drawing zero copies up to the maximum possible. Because the calculation works with log-factorials, it remains precise even for 100-plus-card Commander decks where the raw binomial coefficients would otherwise overflow.

Example and tips

Take a 60-card deck with 4 copies of a key card and a 7-card opening hand. Asking for at least one copy returns about 39.9% — a number worth knowing before you build around that card. Bumping the draw count to model later turns shows how quickly that probability climbs.

Practical pointers:

For “do I draw any of this group”, set k to 1 and choose at-least.
For lands specifically, set K to your land count and n to 7 to see your no-keep risk.
The distribution table is the fastest way to compare, say, running three versus four copies of a card: change K and watch the at-least-one figure move.