Session 5.5.1 – Multinomial Probability Distribution

What is it? The multinomial distribution generalizes the binomial distribution to k categories. We perform n independent trials, and in each trial the outcome falls into exactly one category $1,2,\dots,k$ with fixed probabilities $p_1,\dots,p_k$, where $\sum_{i=1}^k p_i=1$. We want the probability that the counts are $X_1=x_1,\dots,X_k=x_k$ with $\sum_{i=1}^k x_i=n$.

\[ P(X_1=x_1,\dots,X_k=x_k) = \frac{n!}{x_1!\,x_2!\cdots x_k!}\; p_1^{x_1} p_2^{x_2}\cdots p_k^{x_k}. \]

Example (4 classes): Digital bits labeled E, G, F, P

Number of trials n:

#	Label	Probability $p_i$	Count $x_i$	$E[X_i]=n p_i$	$\mathrm{Var}(X_i)=n p_i(1-p_i)$
Sums →		—	—	—	—

Checks: probs sum to 1 → — ; counts sum to n → —

Key properties

Independence across trials; probabilities $p_i$ are fixed for each trial.
All sequences with the same counts $(x_1,\dots,x_k)$ have the same probability. The number of such sequences is $\displaystyle \frac{n!}{x_1!\cdots x_k!}$.
Each $X_i$ is binomial: $\displaystyle X_i \sim \mathrm{Binomial}(n,p_i),\quad E[X_i]=n p_i,\quad \mathrm{Var}(X_i)=n p_i(1-p_i).$
The counts sum to $n$: $\displaystyle \sum_{i=1}^k X_i = n.$ This constraint induces negative correlation among the counts.

Excel steps (copy/paste)

Put labels in A2:A5 (E,G,F,P), probabilities in B2:B5, counts in C2:C5, and n in C6 (or use =SUM(C2:C5)).
Check sums: B6: =SUM(B2:B5) (should be 1), C6: =SUM(C2:C5) (should equal n)
Multinomial coefficient (ways) in D2: =MULTINOMIAL(C2,C3,C4,C5)
Product of $p^x$ in E2: =PRODUCT(B2^C2, B3^C3, B4^C4, B5^C5) Log-safe: =EXP(SUM(C2*LN(B2), C3*LN(B3), C4*LN(B4), C5*LN(B5)))
Final probability in F2: =D2*E2
E[X_i] in D3:D6: =C$6*B2 then drag down
Var(X_i) in E3:E6: =C$6*B2*(1-B2) then drag down

Tip: If B6 ≠ 1, scale probs by =B2/$B$6 etc. If C6 ≠ n, change one count or compute “last x = n − (sum others)”.