Mtg Hypergeometric Calculator

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Module Reviewed by: David Chen, Statistician & MTG Judge

This calculator employs the Hypergeometric Distribution, the mathematically correct model for calculating probabilities in Magic: The Gathering (MTG) decks, ensuring accuracy for deck analysis and optimization.

Optimize your Magic: The Gathering deck building and mulligan decisions. Use the tool below to precisely calculate the probability of drawing a specific number of key cards in your opening hand or during any draw sequence.

MTG Hypergeometric Calculator

Calculated Probability
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MTG Hypergeometric Calculator Formula

P(X=k) = [ C(K, k) * C(N-K, n-k) ] / C(N, n) Formula Source: StatTrek Hypergeometric Distribution Context Source: Math StackExchange (MTG Application)

Variables Explained

  • Total Deck Size ($N$): The total number of cards in your deck (e.g., 60 for Standard, 100 for Commander).
  • Desired Cards in Deck ($K$): The count of the specific card or group of cards you are looking for (e.g., 4 copies of a key spell).
  • Cards Drawn ($n$): The number of cards you are drawing or viewing (e.g., 7 for an opening hand, or 10 if you’ve drawn 3 extra cards).
  • Target Successes ($k$): The exact or minimum number of desired cards you want to see in your drawn sample.

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What is the MTG Hypergeometric Calculator?

The MTG Hypergeometric Calculator applies the Hypergeometric Distribution, a statistical model used to calculate probabilities when drawing without replacement from a finite population. In the context of Magic: The Gathering, this population is your deck. When you draw a card, the probability of drawing any subsequent card changes because that first card is no longer in the deck—this is the “without replacement” condition that the hypergeometric formula handles precisely.

Unlike the simpler Binomial Distribution, the Hypergeometric model is essential for accurate MTG deck analysis. It allows players to quantify the chances of having specific cards (like a certain land count, or a key combo piece) in an opening hand (a 7-card draw) or after a sequence of draws. This calculation is the backbone of informed mulligan decisions and strategic deck construction, letting players optimize card counts for consistency.

By knowing the exact probability of hitting your land drops or drawing your crucial finisher, you move beyond guesswork and into mathematically grounded decision-making, significantly improving your overall win rate.

How to Calculate MTG Hypergeometric Probability (Example)

Imagine you have a 60-card deck and want to draw at least 1 copy of a 4-of card in your 7-card opening hand:

  1. Define Variables: $N=60$ (Total Deck Size), $K=4$ (Desired Copies), $n=7$ (Cards Drawn), $k=1$ (Target Successes).
  2. Calculate Denominator: Find the total number of unique 7-card hands from a 60-card deck: $C(60, 7) = 386,206,920$.
  3. Calculate Numerator (0 Successes): Find the number of hands with exactly 0 desired cards. This is $C(4, 0) \times C(60-4, 7-0) = C(4, 0) \times C(56, 7)$. $C(4, 0)=1$. $C(56, 7)=245,397,800$. Numerator $= 245,397,800$.
  4. Calculate $P(X=0)$: $P(X=0) = 245,397,800 / 386,206,920 \approx 63.54\%$.
  5. Find $P(X \ge 1)$: The probability of drawing at least 1 is $1 – P(X=0)$. $1 – 0.6354 = 0.3646$, or $36.46\%$.

Frequently Asked Questions (FAQ)

Is the Hypergeometric Distribution used for Commander (EDH) decks?

Yes. The formula is universal. You simply adjust the total deck size ($N$) to 99 or 100 (depending on if you count the Commander in the deck), and $K, n, k$ accordingly. The core logic remains the same.

Why can’t I just use the Binomial Distribution?

The Binomial Distribution assumes “sampling with replacement” (the probability stays the same after each draw). Since you don’t shuffle a drawn card back into your MTG deck, the probability of drawing the next card changes, making the Hypergeometric Distribution the only mathematically accurate choice.

What is the maximum $N$ (Total Deck Size) that is supported?

The calculator uses standard JavaScript numbers, which can handle large factorials up to a point using `BigInt` (which is implemented here). For typical MTG decks (up to 200 cards), it is highly accurate. It can handle larger numbers, but precision may decrease for very extreme inputs due to floating-point limits.

How should I use this calculator for mulligan decisions?

Calculate the probability of drawing a “keepable” hand (one with minimum lands and a key early play). If the probability is too low (e.g., under 35-40%), the hand is likely a riskier mulligan, prompting you to ship it for a new, smaller hand.

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