Multinomial Distribution Reference
The multinomial distribution generalizes the binomial distribution to multiple categories. It models the probability of observing specific counts across k categories in n trials.
Probability Formula
P(X₁ = n₁, X₂ = n₂, …, Xₖ = nₖ) = (n! / (n₁! × n₂! × … × nₖ!)) × p₁^n₁ × p₂^n₂ × … × pₖ^nₖ
Where:
- n = total number of trials
- nᵢ = number of outcomes in category i
- pᵢ = probability of category i
- k = number of categories
- n₁ + n₂ + … + nₖ = n
- p₁ + p₂ + … + pₖ = 1
Dice Rolling Examples (6 categories, equal probability)
Rolling a Fair Die n Times
Each outcome has probability p = 1/6
| n rolls | Outcome Pattern | Probability |
|---|---|---|
| 6 | (1,1,1,1,1,1) - each face once | 0.0154 |
| 6 | (2,1,1,1,1,0) - one face twice | 0.2315 |
| 6 | (3,1,1,1,0,0) - one face 3 times | 0.1543 |
| 6 | (2,2,1,1,0,0) - two faces twice | 0.2315 |
| 6 | (6,0,0,0,0,0) - all same face | 0.0002 |
P(all different) when rolling n dice
| n dice | P(all different) |
|---|---|
| 1 | 1.0000 |
| 2 | 0.8333 |
| 3 | 0.5556 |
| 4 | 0.2778 |
| 5 | 0.0926 |
| 6 | 0.0154 |
Trinomial Distribution (k = 3)
n = 5 trials with p₁ = p₂ = p₃ = 1/3
| (n₁, n₂, n₃) | Probability |
|---|---|
| (5, 0, 0) | 0.0041 |
| (4, 1, 0) | 0.0206 |
| (3, 2, 0) | 0.0412 |
| (3, 1, 1) | 0.0823 |
| (2, 2, 1) | 0.1235 |
n = 6 trials with p₁ = 0.5, p₂ = 0.3, p₃ = 0.2
| (n₁, n₂, n₃) | Probability |
|---|---|
| (6, 0, 0) | 0.0156 |
| (5, 1, 0) | 0.0281 |
| (5, 0, 1) | 0.0188 |
| (4, 2, 0) | 0.0211 |
| (4, 1, 1) | 0.0281 |
| (3, 3, 0) | 0.0084 |
| (3, 2, 1) | 0.0337 |
| (2, 2, 2) | 0.0135 |
Blood Type Distribution (US Population)
Blood type probabilities: O = 0.44, A = 0.42, B = 0.10, AB = 0.04
Sample of n = 10 people
| (O, A, B, AB) | Probability |
|---|---|
| (4, 4, 1, 1) | 0.0587 |
| (5, 4, 1, 0) | 0.0391 |
| (4, 4, 2, 0) | 0.0293 |
| (5, 3, 2, 0) | 0.0352 |
| (4, 5, 1, 0) | 0.0373 |
| (4, 4, 0, 2) | 0.0029 |
Expected Values and Covariances
Mean
E[Xᵢ] = n × pᵢ
Variance
Var(Xᵢ) = n × pᵢ × (1 - pᵢ)
Covariance
Cov(Xᵢ, Xⱼ) = -n × pᵢ × pⱼ (for i ≠ j)
Example: n = 100, p₁ = 0.3, p₂ = 0.5, p₃ = 0.2
| Category | E[Xᵢ] | Var(Xᵢ) |
|---|---|---|
| 1 | 30 | 21 |
| 2 | 50 | 25 |
| 3 | 20 | 16 |
Cov(X₁, X₂) = -100 × 0.3 × 0.5 = -15
Multinomial Coefficient Calculator
The multinomial coefficient:
C(n; n₁, n₂, …, nₖ) = n! / (n₁! × n₂! × … × nₖ!)
Common Values
| n | (n₁, n₂, …) | Coefficient |
|---|---|---|
| 3 | (1, 1, 1) | 6 |
| 4 | (2, 1, 1) | 12 |
| 4 | (2, 2) | 6 |
| 5 | (2, 2, 1) | 30 |
| 5 | (3, 1, 1) | 20 |
| 6 | (2, 2, 2) | 90 |
| 6 | (3, 2, 1) | 60 |
| 6 | (4, 1, 1) | 30 |
| 10 | (3, 3, 2, 2) | 25,200 |
| 10 | (4, 3, 2, 1) | 12,600 |
| 10 | (5, 3, 2) | 2,520 |
Chi-Square Goodness of Fit
The multinomial distribution is the basis for chi-square goodness-of-fit tests.
Test Statistic
χ² = Σ (Oᵢ - Eᵢ)² / Eᵢ
Where:
- Oᵢ = observed count in category i
- Eᵢ = expected count = n × pᵢ
Degrees of Freedom
df = k - 1
Example Problems
Example 1: Dice Rolling
Problem: Roll a fair die 10 times. What’s P(exactly 2 ones, 2 twos, 2 threes, and 4 other outcomes)?
Solution:
- n = 10, k = 6
- (n₁, n₂, n₃, n₄, n₅, n₆) where n₁=2, n₂=2, n₃=2, and rest total 4
- Need to sum over all ways the remaining 4 can be distributed
- This requires summing multiple multinomial probabilities
Example 2: Survey Responses
Problem: 30% prefer A, 50% prefer B, 20% prefer C. In 5 surveys, P(2A, 2B, 1C)?
Solution:
- n = 5, (n₁, n₂, n₃) = (2, 2, 1)
- P = (5!/(2!×2!×1!)) × 0.3² × 0.5² × 0.2¹
- P = 30 × 0.09 × 0.25 × 0.2
- P = 0.135
Example 3: Genetics
Problem: Mendel’s peas: 9:3:3:1 ratio expected. In 16 offspring, P(9 round-yellow, 3 round-green, 3 wrinkled-yellow, 1 wrinkled-green)?
Solution:
- p₁ = 9/16, p₂ = 3/16, p₃ = 3/16, p₄ = 1/16
- n = 16, (n₁, n₂, n₃, n₄) = (9, 3, 3, 1)
- P = (16!/(9!×3!×3!×1!)) × (9/16)⁹ × (3/16)³ × (3/16)³ × (1/16)¹
- P ≈ 0.0416
Relationship to Other Distributions
| Relationship | Description |
|---|---|
| Binomial | Multinomial with k=2 |
| Categorical | Single trial (n=1) |
| Dirichlet | Conjugate prior |
When to Use Multinomial
✓ Multiple categories (k > 2)
✓ Fixed number of trials
✓ Categories are mutually exclusive
✓ Probabilities sum to 1
✓ Independent trials
Applications
- Genetics: Phenotype ratios
- Marketing: Brand preferences
- Elections: Vote distributions
- Quality Control: Defect categories
- Surveys: Response distributions
- Biology: Species counts
Related Resources
- Multinomial Lesson - Complete guide
- Binomial Table - Two categories
- Chi-Square Table - Goodness of fit tests
- Probability Calculator - Calculate probabilities