MTG Hypergeometric Calculator
Use this MTG hypergeometric calculator to find your exact draw odds across Commander, Standard, and Limited. Stop guessing land counts or mulligan lines. Plug in your deck parameters to see real card probabilities instantly.
What is the hypergeometric distribution in MTG?
An MTG hypergeometric calculator models card draws without replacement from a finite library. Because each draw removes a card and alters the remaining odds, this discrete statistical distribution gives the exact probability of hitting combo pieces, lands, or removal spells across any sample size.
Standard coin flips rely on binomial distribution, which assumes infinite pools with replacement. In Magic, your library shrinks with every draw step. Hypergeometric modeling accounts for population, trials, and successes directly, giving you dependable deckbuilding data.
Hypergeometric Distribution Calculator Magic the Gathering Formula
The exact probability of finding exactly k target cards in your draw sample is calculated with this formula:
P(X = k) = [ C(K, k) * C(N - K, n - k) ] / C(N, n)N (Deck Size / Library Size): Total cards remaining in the deck (e.g., 99 in Commander, 60 in Standard, 40 in Limited).
K (Successes in Population): Total copies of a card or category in the deck (e.g., 24 lands, 4 Lightning Bolts).
n (Sample Size): Cards drawn / opening hand size (e.g., 7 cards opening hand, 10 cards drawn by turn 4).
k (Observed Successes): Target number of hits you want to draw.
C(a, b): Combinations formula computed as a! / [b! * (a - b)!].
Worked Example: Chance of drawing 2 copies of a card in opening hand
In a 60-card deck (N = 60) with 4 copies of a card (K = 4), drawing an opening hand of 7 (n = 7):
- Exactly 2 copies: P(X = 2) = 8.82%
- At least 1 copy: P(X ≥ 1) = 39.95%
- At least 2 copies: P(X ≥ 2) = 9.68%
How Many Lands Should I Run in a 60 Card Deck?
A standard mana base / land ratio in a 60-card deck typically runs between 23 and 26 lands. Aggro decks with low curves run 20 to 22 lands, midrange lists run 24 to 25 lands, and control decks run 26 to 27 lands to guarantee land drops through turn 5.
Commander (99-Card Library Size): With higher variance, running 36 lands provides a 58.1% chance of hitting 3+ lands in your opening 7. Increasing this to 38 lands raises that rate to 62.9%. Adding 10 two-mana ramp spells provides a 53.6% probability of opening at least one ramp piece.
Limited (40-Card Deck): The standard baseline is 17 lands. This setup gives an 82.2% chance of finding 2 or more lands in an opening hand of 7, protecting you against early mana screw.
Probability of Drawing a Card by Turn 3 MTG
Competitive deck construction requires analyzing cumulative probability (P(X ≥ k)) for draws by turn N:
- Turn 1 (7 cards on play / 8 on draw): 39.9% chance to see a 4-of on the play (44.5% on the draw).
- Turn 3 (9 cards on play / 10 on draw): 48.8% chance to hit a 4-of on the play (53.0% on the draw).
- Turn 4 (10 cards on play / 11 on draw): 53.0% chance on the play (56.9% on the draw).
Under the London Mulligan, you always draw 7 cards and put extras on the bottom. Seeing two independent 7-card hands pushes your cumulative odds of finding at least one key 4-of from 39.95% up to 63.94%.
Frequently Asked Questions
What is the hypergeometric distribution in MTG?
The hypergeometric distribution in MTG calculates the exact probability of drawing a specific number of target cards from a finite deck without replacement. Because drawn cards leave your library, the probability shifts with every single draw.
How many lands do I need for a 40 card deck?
A standard 40-card Limited or Draft deck runs 17 lands. This land ratio gives an 82.2% chance of opening at least 2 lands in a 7-card hand and roughly a 54.3% chance of drawing 3 or more lands by turn 3.
What are the odds of a mulligan improving my hand?
Under London Mulligan rules, you draw a fresh 7 cards. For a key 4-of card in a 60-card deck, your chance of seeing at least one copy in the opening hand rises from 39.95% on the first draw to 63.94% across two 7-card attempts.
How do you calculate draw probability in Magic the Gathering?
You calculate MTG draw odds using the hypergeometric formula: P(X = k) = [C(K, k) * C(N - K, n - k)] / C(N, n), where N is library size, K is card copies, n is sample size drawn, and k is target successes.
MTG Probability Simulator
Cumulative Probability: P(X ≥ 3)
At Least 3 HitsExact (3)
29.12%
At Most (≤3)
76.65%
Zero Hits (0)
3.30%
Turn-by-Turn Mana Curve
Chance of drawing ≥ 3 targets by turn
| Turn | Cards Seen | Probability | Visual |
|---|---|---|---|
| T1 | 7 cards | 52.5% | |
| T2 | 8 cards | 63.5% | |
| T3 | 9 cards | 72.7% | |
| T4 | 10 cards | 80.0% | |
| T5 | 11 cards | 85.7% | |
| T6 | 12 cards | 90.0% | |
| T7 | 13 cards | 93.1% |
London Mulligan Hit Simulator
Probability of finding ≥ 3 target(s) across consecutive 7-card hands
Hand 1 (Keep 7)
52.5%
Mull to 6 (14 seen)
77.4%
Mull to 5 (21 seen)
89.3%
* Probability calculations use the hypergeometric distribution without replacement. Real-world game variance, scry decisions, shuffle randomness, and deck-thinning fetch mechanics will dynamically alter draw outcomes during live play.
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