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What is hypergeometric distribution?

Hypergeometric distribution is a probability distribution that describes the likelihood of drawing a specific number of success cards from a finite population without replacement. Unlike binomial distribution (which assumes replacement), hypergeometric distribution perfectly models the act of drawing cards from an MTG deck.

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The key principle

When you draw a card from your deck, you cannot draw that same card again. This “sampling without replacement” is exactly what hypergeometric distribution was designed to model, making it the ideal statistical tool for MTG probability calculations.

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Why it’s perfect for MTG

Magic: The Gathering deck analysis has unique characteristics:
  • Fixed population: Your deck has exactly 60 cards (or 99 in Commander)
  • No replacement: Once you draw a card, it’s no longer in your deck
  • Known quantities: You know exactly how many lands and spells are in your deck
  • Specific outcomes: You want to calculate the probability of drawing specific combinations (e.g., “3 lands by turn 3”)
These constraints make hypergeometric distribution the mathematically correct approach, as opposed to other probability methods that assume infinite populations or replacement.

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The formula

The hypergeometric probability formula is: 
P(X = k) = [C(K, k) × C(N-K, n-k)] / C(N, n)
Where:
  • N = Total population size (deck size, typically 60)
  • K = Number of success states in population (e.g., number of lands in deck)
  • n = Number of draws (e.g., 7 cards for opening hand)
  • k = Number of observed successes (e.g., exactly 3 lands drawn)
  • C(a, b) = Binomial coefficient “a choose b” = a! / (b! × (a-b)!)

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Breaking down each variable

The total number of cards in your deck. For Standard and Modern, this is typically 60 cards. For Commander, it’s 99 cards plus your commander.Example: If you have a 60-card deck, N = 60.
The total number of cards in your deck that meet your criteria. If you’re calculating land probability, K is the number of lands in your deck.Example: If your deck has 24 lands, K = 24.
How many cards you’re drawing. For an opening hand, this is 7. For “by turn 3 on the play,” this would be 9 (7 opening + 2 draw steps).Example: Opening hand is 7 cards, so n = 7.
The specific number of success cards you want to draw. If you want exactly 3 lands, k = 3.Example: To have exactly 3 lands in your opening hand, k = 3.

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Frank Karsten’s pioneering work

This implementation is based on the groundbreaking analysis by Frank Karsten, a Hall of Fame player and statistician. His ChannelFireball article “How Many Colored Mana Sources Do You Need to Consistently Cast Your Spells?” established the mathematical framework for modern MTG mana base construction.
Karsten’s work showed that you need approximately 14 sources of a color to cast a 1-mana spell on turn 1 with 90% consistency, and 19 sources for a 2-mana spell on turn 2.

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How it differs from other methods

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vs. Binomial distribution

Binomial assumes:
  • Infinite population (or replacement after each draw)
  • Each trial is independent
Hypergeometric handles:
  • Finite population (your 60-card deck)
  • Dependent trials (drawing one card affects the next)

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vs. Normal approximation

Normal approximation is useful for:
  • Very large populations
  • Rough estimates
Hypergeometric provides:
  • Exact probabilities
  • Accurate results for small populations like MTG decks

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vs. Simulation (Monte Carlo)

Monte Carlo simulation:
  • Approximate through repeated random trials
  • Computationally expensive
  • Requires many iterations for accuracy
Hypergeometric:
  • Exact mathematical calculation
  • Computationally efficient
  • Single calculation provides precise probability

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Visual examples

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Example 1: Simple land calculation

Deck: 60 cards
Lands: 24
Spells: 36

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Example 2: Cumulative probability

Often you don’t want “exactly” 3 lands—you want “at least” 3 lands (or “at most” 4 lands). 
P(X ≥ 3) = P(X = 3) + P(X = 4) + P(X = 5) + P(X = 6) + P(X = 7)
This is closer to real gameplay questions like “What’s the chance I have a playable hand?”

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Example 3: Multi-color analysis

For colored mana sources, you need to consider:
  • How many sources produce the required color
  • Whether dual lands can satisfy multiple requirements
  • The converted mana cost needed by a certain turn
10 Island (produces {U})
10 Mountain (produces {R})
4 Steam Vents (produces {U} or {R})
36 spells

Card to cast: Lightning Bolt {R}
The MTG Deck Builder calculator handles all these complex scenarios automatically, including fetch lands, dual lands, and multi-color requirements.

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Key takeaways

  1. Hypergeometric distribution models MTG perfectly because decks are finite populations with sampling without replacement
  2. The formula calculates exact probabilities, not approximations
  3. Frank Karsten’s work established the gold standard for mana base analysis using this method
  4. It’s superior to other probability methods for card games with finite, known deck compositions
  5. The calculator implements this formula to analyze every card in your deck against every possible hand combination

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Next steps

How it works

Learn how the algorithm implements hypergeometric distribution

Examples

See real-world examples with different deck types

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