Multinomial Distribution Calculator

Multinomial Distribution Formula:

\[ P = \frac{n!}{k_1! k_2! \cdots k_m!} \times p_1^{k_1} \times p_2^{k_2} \times \cdots \times p_m^{k_m} \]

Multinomial Probability (P):

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1. What is the Multinomial Distribution?

The multinomial distribution is a generalization of the binomial distribution that models the probability of counts for each possible outcome when sampling with replacement from a population with multiple categories.

2. How Does the Calculator Work?

The calculator uses the multinomial probability formula:

\[ P = \frac{n!}{k_1! k_2! \cdots k_m!} \times p_1^{k_1} \times p_2^{k_2} \times \cdots \times p_m^{k_m} \]

Where:

\( n \) — Total number of trials
\( k_i \) — Number of times outcome \( i \) occurs
\( p_i \) — Probability of outcome \( i \)

Explanation: The formula calculates the probability of observing a specific combination of counts across multiple categories.

3. Applications of Multinomial Distribution

Details: Used in genetics (genotype frequencies), marketing (customer choice modeling), natural language processing (text classification), and more.

4. Using the Calculator

Tips: Enter total trials (n), comma-separated outcome counts (k values), and comma-separated probabilities (p values). Probabilities must sum to 1 and counts must sum to n.

5. Frequently Asked Questions (FAQ)

Q1: How is this different from binomial distribution?
A: Binomial has two outcomes (success/failure), multinomial extends this to multiple categories.

Q2: What are the requirements for valid inputs?
A: n must be positive integer, k values must be non-negative integers summing to n, p values must be non-negative and sum to 1.

Q3: When would I use this distribution?
A: When modeling scenarios with multiple possible outcomes and fixed probabilities for each.

Q4: What if my probabilities don't sum to exactly 1?
A: The calculator will show an error as this violates probability rules.

Q5: Can I use decimal values for counts?
A: No, counts must be whole numbers as they represent actual occurrences.