Hypergeometric Calculator: Card Draw Odds for TCG Decks

1. Enter the deck size and the number of copies of the card or card type you want. Use a chip for a common deck size.
2. Enter the cards drawn, which is your opening hand plus every draw so far, and the copies you want at least.
3. Optionally fill in group B for a second need, add mulligans, and read the turn table below the result.

Here is what each result means:

The hypergeometric distribution gives the probability of a number of successes when you draw from a finite set without putting cards back. A shuffled deck is the standard example.

It differs from a coin flip because each card you draw changes what is left. Draw a land and the deck holds fewer lands, so the next card is a little less likely to be one.

Card players use it for opening hands, land counts and "do I draw my combo piece by turn four". Statistics classes use it for lots and quality sampling.

This tool covers one or two card groups. It does not know what a card does, so you decide which cards belong in each group.

The calculator counts the hands that contain exactly i wanted cards, divides by all possible hands, and adds up the cases you care about.

1. Count the ways to pick i wanted cards from the K copies: C(K, i).
2. Count the ways to fill the rest of the hand from the other N - K cards: C(N - K, n - i).
3. Divide the product by every possible hand, C(N, n), to get P(X = i).
4. Add P(X = i) for every i you accept. For "at least k", add from k up to the most you can hold.
5. For two groups, sum over pairs (i, j) of C(K1, i) x C(K2, j) x C(rest, n - i - j) divided by C(N, n).

The math uses logarithms of factorials, so a deck of 1,000 cards does not overflow. Exact fractions in a reference script agree with the calculator.

Take a 60-card deck with 24 copies of a card type, such as lands. You look at 7 cards and want at least 3 of them.

Calculation: add the chances of exactly 3, 4, 5, 6 and 7 copies. Each chance is C(24, i) x C(36, 7 - i) / C(60, 7). The sum is 58.79%. The chance of exactly 3 is 30.87%, and the chance of 2 or fewer is 41.21%.

The expected number of copies in 7 cards is 7 x 24 / 60 = 2.8. So you see fewer than 3 copies about 41.21% of the time, even with this many.

If you allow a mulligan, each new seven-card hand is a fresh shuffle. The chance that at least one of two hands has 3 or more copies is 1 - (1 - 0.5879)^2 = 83.02%, and for three hands it is 93.00%.

For two groups at once, take 24 lands and 8 two-drops in 60 cards after 9 cards. The chance of at least 2 lands and at least 1 two-drop together is 70.11%.

Five inputs decide the answer. Small changes move it more than most players expect.

Deck Size

A bigger deck dilutes every card. Keeping 24 copies but moving from 60 to 100 cards cuts the expected count in seven cards from 2.80 to 1.68.

Copies in the Deck

Each added copy helps less than the one before it. Going from 4 to 8 copies matters more than going from 20 to 24.

Cards Seen

Every extra card seen adds to the chance. This is why the turn table matters: turn 3 on the play is 9 cards seen, and turn 3 on the draw is 10.

How Many Copies You Need

Asking for 3 copies is much harder than asking for 1. Use "at least" for goals and "exactly" only for rare questions.

Mulligans and Card Selection

A mulligan reshuffles the deck, so each new hand is a separate trial. Cards that dig deeper, such as card draw spells, raise cards seen and are not modelled one by one.

Many calculators use the binomial model, which treats every draw as independent with the same chance. That model is wrong for a deck because cards are not replaced.

The binomial figure is lower here because it allows the same copy to show up twice. The gap grows when the deck is small or when you draw many cards. The values in the table were computed with the same code as the tool.

The chips use sizes that appear in published rules. Other games and house rules differ, so check your own rulebook.

The table lists sizes from the Magic Comprehensive Rules effective 25 September 2026 and the Pokemon TCG tournament handbook revised 1 September 2026. Both were read on 2026-10-03. Rules and formats change, so confirm the current text before an event.

Choosing a Land or Energy Count

Set group A to your land or energy count and ask for a number you need by a turn. Compare 22, 24 and 26 and watch how the odds move.

Checking a Key Card

Set copies to 1, 2, 3 or 4 and read the turn table. It shows the turn by which you can expect to have found it at a given confidence.

Planning Two Needs at Once

Use group B to ask for lands and early plays together. The joint chance is always lower than either one alone.

Comparing Dice and Card Odds

Dice rolls are independent. Cards are not. For repeated independent tries, such as a loot drop, read understanding loot drop odds.

1. Counting the Opening Hand Wrong

Cards drawn means every card you have seen. On the play you skip the first draw, so turn 3 is the opening hand plus 2 draws.

2. Using the Binomial Model

It overstates or understates results because it ignores that cards leave the deck.

3. Mixing Up at Least and Exactly

"Exactly 3 lands" is not the same as "3 or more lands". Most goals are "at least".

4. Forgetting Overlapping Groups

A card in both groups counts twice. Group A and group B in this tool must not overlap, and their total cannot be more than the deck size.

5. Treating Odds as a Promise

A 90 percent chance still fails one time in ten. Keep a plan for the bad draw.

The math is exact for a fully shuffled deck, and the answer is only as good as your inputs.

The mulligan result treats each new hand as an independent shuffle. It tells you whether a hand meets the goal, not what you keep.

To see the idea behind "chance versus odds", read probability vs odds. For counting hands directly, read combinations vs permutations, and for averages read expected value explained.

Use the hypergeometric formula: choose the wanted cards, choose the rest of the hand from the other cards, and divide by all possible hands. For one 4-of in a 60-card deck, the chance of at least 1 copy in 7 cards is 39.95 percent.

With 24 lands in a 60-card deck, it is 58.79 percent in 7 cards. With 17 lands in a 40-card deck it is 64.93 percent. Enter your own deck for your numbers.

The hypergeometric model removes each drawn card from the deck, so odds change with every draw. The binomial model keeps the chance fixed, which suits dice but not decks.

Use your real deck size. Common sizes are 40, 60, 99 and 100, and the chips set them. Magic constructed needs at least 60, limited at least 40, and Commander uses 100 including the commander.

Use 99 when the commander is already in the command zone and you draw from the library. Use 100 only if you want to treat the commander as part of the random deck, which does not happen in play.

On the play you skip the first draw, so turn 1 is your opening hand. On the draw you take one card on turn 1. The turn table follows that rule for you.

Each new hand is a fresh shuffle, so the chance that at least one of several hands works is 1 - (1 - p)^hands. With p of 58.79 percent, two hands give 83.02 percent.

Yes. Fill in group B for a second type. The tool then shows the chance of meeting both goals in the same cards, which is always lower than either alone.

The average is cards drawn times copies divided by deck size, such as 2.8 lands in 7 cards. You cannot see 2.8 cards, so the chance of reaching 3 is a separate number.

Yes, for any game that draws from a shuffled deck without replacement. Enter that deck size and copies. Check your game rules for deck limits, because this page only cites Magic and the Pokemon TCG.

• Hypergeometric Distribution (OpenStax) (Introductory Statistics 4.5, Rice University). Definition and formula.
• Hypergeometric distribution (Wikipedia) (Wikipedia). Probability mass function, cumulative sums and comparison with the binomial.
• Magic: The Gathering Comprehensive Rules (Wizards of the Coast, effective 25 September 2026). Rules 100.2a, 100.2b, 103.5, 903.5a and 903.6.
• Pokemon TCG Tournament Handbook (Play! Pokemon, revision 1 September 2026). Constructed decks have exactly 60 cards.