Optimize your Magic: The Gathering deck by calculating the exact probability of drawing a minimum number of lands in your opening hand. Ensure you are never mana-screwed or mana-flooded again!
Magic: The Gathering Land Calculator
Probability of drawing at least N lands:
Magic: The Gathering Land Calculator Formula
The probability $P(X \ge N)$ of drawing $N$ or more lands is the sum of probabilities of drawing exactly $N$ lands up to $H$ lands:
$$P(X \ge N) = \sum_{k=N}^{H} P(X=k)$$
$$P(X=k) = \frac{\binom{L}{k} \binom{D-L}{H-k}}{\binom{D}{H}}$$
Where $\binom{n}{k} = \frac{n!}{k!(n-k)!}$ is the binomial coefficient (combinations).
Formula Source: Math StackExchange (Hypergeometric Distribution), MTG Fandom Wiki (Probability)
Variables
- Total Deck Size (D): The total number of cards in your deck (usually 60 in Standard/Modern).
- Total Lands in Deck (L): The number of basic and non-basic land cards included in your deck.
- Opening Hand Size (H): The number of cards drawn in the initial hand (usually 7).
- Minimum Target Lands (N): The minimum number of lands you want to see in your opening hand (e.g., 2 or 3).
Related Calculators
What is Magic: The Gathering Land Calculator?
The MTG Land Calculator is an essential tool for competitive and casual players alike, allowing them to precisely determine the mathematical consistency of their deck’s mana base. By applying the principles of the Hypergeometric Distribution, the calculator assesses the statistical likelihood of having a playable hand—one that contains an adequate number of lands to cast spells in the early game.
Since Magic: The Gathering is a game of probability, relying on intuition alone is insufficient. This calculator helps you fine-tune the ratio of land to non-land cards, often between 22 and 26 lands in a 60-card deck, depending on the average converted mana cost (CMC) of your spells. A low probability of drawing 2-3 lands suggests a high risk of ‘mana screw,’ while a very high probability suggests ‘mana flood.’
How to Calculate Land Probability (Example)
- Define Inputs: Assume a 60-card deck (D=60), 24 lands (L=24), 7-card hand (H=7), and a desire for at least 3 lands (N=3).
- Calculate Denominator: Determine the total number of ways to draw a 7-card hand from 60 cards: $\binom{60}{7}$.
- Calculate Individual Probabilities: Calculate the probability of drawing exactly 3, 4, 5, 6, and 7 lands using the formula $\frac{\binom{L}{k} \binom{D-L}{H-k}}{\binom{D}{H}}$ for each $k$.
- Sum the Results: Add the probabilities for $P(X=3) + P(X=4) + P(X=5) + P(X=6) + P(X=7)$.
- Analyze the Result: If the sum is 75%, for instance, you have a 75% chance of getting 3 or more lands, indicating a robust mana base.
Frequently Asked Questions (FAQ)
Is the Hypergeometric Distribution accurate for Magic: The Gathering?
Yes, the Hypergeometric Distribution is the mathematically correct model for calculating probabilities in Magic: The Gathering (or any card game) when drawing without replacement from a finite pool of cards, such as an opening hand.
What is the ideal number of lands in a 60-card deck?
There is no single ideal number, but most competitive 60-card decks use between 22 and 26 lands. Aggro decks with low average mana costs use fewer (20-22), while control decks with high costs use more (25-27).
Does this calculator account for mulligans?
No, this calculation only provides the probability for the initial 7-card draw. The complexity of mulligan decisions (e.g., Scry effects, taking one less card) requires a separate, more complex sequential probability model.
Can I use this calculator for Commander (EDH)?
Yes, but you must manually change the ‘Total Deck Size (D)’ to 99 (since the Commander is outside the deck) and adjust the ‘Opening Hand Size (H)’ accordingly. The fundamental probability model remains the same.