Another important property of the multinomial distribution is that its conditional distributions are also multinomial.

Conditional distributions

If \((X_1, X_2,\dots, X_g)\) have a \(\MultinomDistn(n, \pi_1, \dots, \pi_g)\) distribution, the conditional distribution of \((X_2, \dots, X_g)\), given that \(X_1 = x_1\) is \(\MultinomDistn(n-x_1, \pi_2^*, \dots, \pi_g^*)\) where

\[ \pi_i^* \;\;=\;\; \frac{\pi_i}{1 -\pi_1} \]

The marginal probability function of \(X_1\) is

\[ p_{X_1}(x_1) \;=\; \frac{n!}{x_1!(n - x_1)!} \pi_1^{x_1}(1-\pi_1)^{n-\pi_1} \]

The conditional probabilities for \(X_2, \dots, X_g\), given \(X_1 = x_1\) are therefore

\[ \begin{align} p_{X_2, \dots, X_g \mid X_1=x_1}(x_2, \dots, x_g) \;&=\; \frac{p(x_1, \dots, x_g)}{p_{X_1}(x_1)} \\[0.5em] &=\; \frac{\frac{\large n!}{\large x_1!\;x_2!\; \cdots,\;x_g!} \pi_1^{x_1}\pi_2^{x_2}\cdots \pi_g^{x_g}}{\frac{\large n!}{\large x_1!(n - x_1)!} \pi_1^{x_1}(1-\pi_1)^{n-\pi_1}} \\[0.5em] &=\; \frac{(n - x_1)!}{x_2!\dots x_g!} \left(\frac{\pi_2}{1 - \pi_1} \right)^{x_2} \cdots \left(\frac{\pi_g}{1 - \pi_1} \right)^{x_g} \end{align} \]

and this is the joint probability function of the \(\MultinomDistn(n-x_1, \pi_2^*, \dots, \pi_g^*)\) distribution.

A similar result holds for the conditional distribution of any collection of these multinomial variables, given the values of the others.

Opinion poll

We again consider results from an opinion poll in which responses to a question about a new piece of legislation have probabilities

P(Agree) = 0.3,
P(Neutral) = 0.4
P(Disagree) = 0.3

If a random sample of \(n\) = 4 people is asked and the numbers agreeing, neutral and disagreeing are \(X_A\), \(X_N\) and \(X_D\), then these variables have a \(\MultinomDistn(4, 0.3, 0.4, 0.3)\) distribution.

If it is known that one person in the sample Agrees, i.e. \(X_A = 1\), then there are \(n-x_A = 3\) remaining who are either Neutral or Disagree. The probabilities of these are in proportion to their initial probabilities,

\[ P(Neutral \mid \text{not } Agree) = \frac{0.4}{0.4 + 0.3} \spaced{and} P(Disagree \mid \text{not } Agree) = \frac{0.3}{0.4 + 0.3} \]

The conditional distribution for the numbers who are Neutral and Disagree, given that 1 of the 4 Agrees, is therefore

\[ (X_N, X_D) \;\sim\; \MultinomDistn\left(3, \frac{0.4}{0.4 + 0.3}, \frac{0.3}{0.4 + 0.3}\right) \]

Since this multinomial distribution only involves two variables, it is actually binomial, so the conditional distribution for the number who are Neutral, given that \(X_A = 1\) agrees is

\[ X_N \;\sim\; \BinomDistn\left(n=3, \pi=\frac{0.4}{0.4 + 0.3}\right) \]

Graphical illustration

The following diagram illustrates this result.

The diagram initially shows the joint probabilities for a sample of size 1. First change the sample size to \(n = 4\).

Now click Conditional for Y to display the conditional distributions for the number who are Neutral, given that different numbers agree. This simply scales the joint probabilities for each value of Y to sum to one.

Now click Restrict to and use the slider to show only the conditional distribution when the number agreeing is X=1. This is the \(\BinomDistn\left(n=3, \pi=\frac{0.4}{0.4 + 0.3}\right)\) distribution that was described above.