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Repeated trials and the multinomial model

When each independent trial falls into one of several categories, the multinomial coefficient counts the arrangements and the probabilities multiply.

The formulas

Multinomial probability: $P (N_{1} = k_{1}, \dots, N_{m} = k_{m}) = \frac{n !}{k _{1} ! \dots k _{m} !} p_{1}^{k_{1}} \dots p_{m}^{k_{m}}$
Marginals: $N_{j} \sim Binomial (n, p_{j})$
Covariance: $Cov (N_{i}, N_{j}) = - n p_{i} p_{j}, i \neq = j$
Constraint: $j \sum k_{j} = n, j \sum p_{j} = 1$

Where it comes from

Any one arrangement with those category counts has probability $p_{1}^{k_{1}} \dots p_{m}^{k_{m}}$ by independence.
The number of such arrangements is the multinomial coefficient, so multiply.
Collapsing all categories but one back into 'other' turns the multinomial into a binomial, which gives the marginals immediately.

Worked example

The letters of the word BALANCE are arranged at random. How many distinct arrangements are there?

BALANCE has 7 letters, of which A appears twice and every other letter once.
Distinct arrangements of $n$ items with repeat counts $k_{1}, \dots, k_{m}$ number $\frac{n !}{k _{1} ! \dots k _{m} !}$ — the multinomial coefficient.
Only A repeats, so the denominator is $2!$ and everything else contributes $1!$ .
$\frac{7 !}{2 !} = \frac{5040}{2} = 2, 520$ distinct arrangements.

Answer: 2,520

This answer is recomputed from the site’s own interest-theory and probability functions every time the test suite runs, so the page and the mathematics cannot drift apart.

Memory hooks

The multinomial coefficient is 'total factorial over the product of the repeat factorials'.
Multinomial marginals are binomial. If a question asks about ONE category, forget the rest.

Traps

Forgetting that multinomial counts are negatively correlated — more of one category means fewer of another.
Miscounting the repeats when arranging letters.

Drill this: the Exam P question bank has original questions on this topic, and today’s free round is open to everyone.

Repeated trials and the multinomial model

The formulas

Where it comes from

Worked example

Memory hooks

Traps

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