Hypergeometric Distribution Calculator Mtg

Hypergeometric Formula:

\[ P = \frac{\binom{K}{k} \times \binom{N-K}{n-k}}{\binom{N}{n}} \]

Population Size (N):

cards

Successes in Population (K):

cards

Sample Size (n):

cards

Desired Successes (k):

cards

Exact Probability (P):

Cumulative (≥k):

Cumulative (≤k):

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1. What is Hypergeometric Distribution?

The hypergeometric distribution describes the probability of drawing k successes (desired cards) in n draws (sample size) from a finite population N without replacement, where there are K successes in the population. It's essential for calculating card draw probabilities in games like Magic: The Gathering.

2. How Does the Calculator Work?

The calculator uses the hypergeometric formula:

\[ P = \frac{\binom{K}{k} \times \binom{N-K}{n-k}}{\binom{N}{n}} \]

Where:

\( N \) — Total population size (deck size)
\( K \) — Number of successes in population (copies of desired card)
\( n \) — Sample size (number of cards drawn)
\( k \) — Number of observed successes (desired outcome)

Explanation: The formula calculates the exact probability of drawing exactly k copies of a card when drawing n cards from a deck of size N containing K copies of that card.

3. Importance for MTG Players

Details: Understanding these probabilities helps players optimize their decks, determine optimal land counts, and evaluate the consistency of their strategies. It's fundamental for competitive deck building.

4. Using the Calculator

Tips:

Population (N): Typically 60 for constructed or 40 for limited
Successes (K): Number of copies of the card in your deck (usually 1-4)
Sample (n): Number of cards you'll draw (7 for opening hand, more for later turns)
Desired (k): How many copies you want to draw

5. Frequently Asked Questions (FAQ)

Q1: Why use hypergeometric instead of binomial?
A: MTG involves drawing without replacement from a finite deck, making hypergeometric the correct model. Binomial assumes replacement.

Q2: What's a good probability for key cards?
A: Competitive decks often aim for ≥60% chance to have key cards when needed. Mulligan decisions affect this.

Q3: How does mulliganing affect probabilities?
A: Each mulligan is essentially another independent sample. The calculator shows single-draw probabilities.

Q4: What about card filtering/tutoring?
A: These effects change the calculation. This calculator assumes pure random draws.

Q5: How to calculate probability by turn X?
A: For turn X, sample size is 7 + X - 1 (draw each turn). Remember to account for play/draw.