Multinomial Distribution: Theory, Applications, and a Real-World Example

Introduction

The multinomial distribution is an extension of the binomial distribution, where an experiment consists of n independent trials, and each trial results in one of k mutually exclusive categories. This distribution is widely used in fields such as genetics, marketing, polling, and natural language processing.

In this post, we will explore:

• The theoretical foundation of the multinomial distribution.
• Real-world applications where it is used.
• A numerical example with step-by-step calculations.

Theory of Multinomial Distribution

Definition

If an experiment consists of n independent trials, where each trial results in one of k outcomes with probabilities $p_1, p_2, ..., p_k$, then the number of occurrences of each category follows a multinomial distribution:

$$(X_1, X_2, ..., X_k) \sim \text{Multinomial}(n, p_1, p_2, ..., p_k)$$

where:

• $X_i$ represents the count of category $i$.
• $p_i$ is the probability of category $i$ ($\sum p_i = 1$).
• $X_1 + X_2 + ... + X_k = n$.

Probability Mass Function (PMF)

The probability of obtaining $X_1 = x_1, X_2 = x_2, ..., X_k = x_k$ is:

$$P(X_1 = x_1, X_2 = x_2, ..., X_k = x_k) = \frac{n!}{x_1! x_2! ... x_k!} p_1^{x_1} p_2^{x_2} ... p_k^{x_k}$$

subject to $x_1 + x_2 + ... + x_k = n$.

Mean and Variance

• Expected value: $E(X_i) = n p_i$
• Variance: $\text{Var}(X_i) = n p_i (1 - p_i)$
• Covariance: $\text{Cov}(X_i, X_j) = - n p_i p_j$, for $i \neq j$.

Real-World Applications

1. Genetics (Mendelian Inheritance)

In a monohybrid cross, genotypes follow a 3:1 ratio, which can be modeled as a multinomial distribution.

2. Customer Preference Survey

A company surveys 500 customers about their preferred product among three brands. The number of votes follows a multinomial distribution.

3. Election Polling

If 1,000 voters select from four political parties, the vote counts for each party follow a multinomial model.

4. Dice Rolling (Gaming)

Rolling a fair six-sided die 10 times leads to a multinomial distribution where $p_i = 1/6$ for each face.

Example:

Problem Statement

A company surveys 10 customers about their preferred mobile brand:

• Brand A: $p_1 = 0.4$
• Brand B: $p_2 = 0.35$
• Brand C: $p_3 = 0.25$

Find the probability that:

• 4 customers prefer Brand A,
• 3 customers prefer Brand B,
• 3 customers prefer Brand C.

Solution

Using the multinomial probability formula:

$$P(X_1 = 4, X_2 = 3, X_3 = 3) = \frac{10!}{4!3!3!} (0.4)^4 (0.35)^3 (0.25)^3$$

Computing step-by-step:

$$\frac{10!}{4!3!3!} = 420$$

$$(0.4)^4 = 0.0256, \quad (0.35)^3 = 0.042875, \quad (0.25)^3 = 0.015625$$

Multiplying all terms:

$$420 \times 0.0256 \times 0.042875 \times 0.015625 \approx 0.087$$

Final Answer

The probability that exactly 4 customers choose Brand A, 3 choose Brand B, and 3 choose Brand C is 0.087 (8.7%).

Conclusion

The multinomial distribution is an essential tool for modeling multi-category outcomes.

• The multinomial distribution extends the binomial model to multiple categories.

• It is widely used in genetics, marketing, elections, and text analysis.

• The probability mass function (PMF) helps compute specific event probabilities.

• A real-world example showed how to calculate multinomial probabilities for customer preferences.