The Multinomial Distribution is a generalization of the binomial distribution. While a binomial distribution deals with the probability of success/failure over trials, a multinomial distribution deals with more than two possible outcomes — like rolling a die or choosing a color.

With NumPy, it's simple to simulate and work with multinomial outcomes for experiments, games, and probability modeling.

What is a Multinomial Distribution?

The multinomial distribution describes the probability of counts of multiple outcomes from a fixed number of independent trials, where each trial has more than two possible outcomes.

Probability Mass Function (PMF)

P(X1=x1,...,Xk=xk)=n!x1!⋅x2!⋅⋯⋅xk!⋅p1x1⋅p2x2⋯pkxkP(X_1 = x_1, ..., X_k = x_k) = \frac{n!}{x_1! \cdot x_2! \cdot \dots \cdot x_k!} \cdot p_1^{x_1} \cdot p_2^{x_2} \cdots p_k^{x_k}

Where:

• nn is the total number of trials

• xix_i is the number of times outcome ii occurred

• pip_i is the probability of outcome ii

• ∑xi=n\sum x_i = n and ∑pi=1\sum p_i = 1

Real-Life Examples

NumPy's multinomial() Function

NumPy allows you to sample from a multinomial distribution using:

numpy.random.Generator.multinomial(n, pvals, size=None)

Parameters

✅ Returns

An array of counts of outcomes.

✅ Example: Simulate Rolling a Die

import numpy as np

rng = np.random.default_rng(seed=42)

# Roll a fair 6-sided die 10 times
outcomes = rng.multinomial(n=10, pvals=[1/6]*6)
print("Die roll outcome counts:", outcomes)

Each element in outcomes represents how many times each face appeared.

Visualizing the Results

import matplotlib.pyplot as plt

labels = ['1', '2', '3', '4', '5', '6']
plt.bar(labels, outcomes, color='skyblue')
plt.title("Die Roll Simulation (10 Trials)")
plt.xlabel("Die Face")
plt.ylabel("Frequency")
plt.grid(True, axis='y')
plt.show()

Multiple Simulations

You can simulate this experiment multiple times using the size parameter:

results = rng.multinomial(n=10, pvals=[1/6]*6, size=1000)
print("Shape:", results.shape)  # (1000, 6)

This gives you a 1000x6 matrix — each row is a single 10-trial simulation.

Plot Average Frequencies

avg_outcomes = results.mean(axis=0)

plt.bar(labels, avg_outcomes, color='orange')
plt.title("Average Die Frequencies (1000 Simulations)")
plt.xlabel("Die Face")
plt.ylabel("Average Count per 10 Rolls")
plt.grid(True)
plt.show()

Each bar should approach 10 × (1/6) = 1.67 if the die is fair.

Another Example: Voting Poll

Let’s say an election has 3 candidates with these probabilities:

• Alice: 50%

• Bob: 30%

• Charlie: 20%

We survey 100 voters:

votes = rng.multinomial(n=100, pvals=[0.5, 0.3, 0.2])
candidates = ['Alice', 'Bob', 'Charlie']

plt.bar(candidates, votes, color='green')
plt.title("Simulated Votes for Candidates (n=100)")
plt.ylabel("Votes")
plt.show()

print(dict(zip(candidates, votes)))

Full Simulation: Candy Bag Problem

Imagine a bag of candies with colors:

• Red: 40%

• Green: 35%

• Blue: 25%

Let’s simulate opening 500 candy bags, each containing 20 candies.

colors = ['Red', 'Green', 'Blue']
probs = [0.4, 0.35, 0.25]

# Simulate
bags = rng.multinomial(n=20, pvals=probs, size=500)

# Average count per color
avg_counts = bags.mean(axis=0)

plt.bar(colors, avg_counts, color=['red', 'green', 'blue'])
plt.title("Average Candies per Color (500 Bags)")
plt.ylabel("Average Count")
plt.grid(True)
plt.show()

print("Expected average per bag:", [p*20 for p in probs])
print("Simulated average per bag:", avg_counts.round(2))

Tips

⚠️ Common Pitfalls

Multinomial vs Binomial vs Categorical

Conclusion

The multinomial distribution is essential for simulating and modeling multi-category outcomes over repeated trials. NumPy’s multinomial() function makes it easy to:

• Simulate dice, polls, or surveys

• Run multiple trials at scale

• Analyze outcome distributions

Summary