Multinomial Distribution Calculator: Compute Multi-Outcome Probabilities

The probability mass function (PMF) for the Multinomial Distribution is given by:

P(X₁=k₁, ..., X_m=k_m) = n! ∏_i=1^m k_i!   ×   ∏_i=1^m p_i^k_i

Where:

• n: The total number of independent trials.
• m: The number of distinct categories or outcomes (implied by the number of k_i/p_i values).
• k_i: The number of times outcome i occurs in n trials.
• p_i: The probability of outcome i occurring in any single trial.

Constraints:

• The sum of all outcome counts must equal the total number of trials: ∑_i=1^m k_i = n
• The sum of all outcome probabilities must equal 1: ∑_i=1^m p_i = 1 (within a small tolerance for floating point numbers).
• 0 ≤ p_i ≤ 1 for all i.
• 0 ≤ k_i ≤ n for all i, and k_i must be integers.

This formula determines the probability of observing a specific distribution of counts (k₁, ..., k_m) across multiple categories, given a total number of trials (n) and the individual probabilities (p₁, ..., p_m) for each category.