The Dirichlet distribution

Is it to proportions what Multinomial is to number in categories?

Interpretation

Let us consider an urn with $K$ different colors. There are ${\alpha }_1$ balls of the first colors, …, ${\alpha }_K$ balls of the $K$th color. We draw $N$ balls from this urn using the following protocol:

1. Pick one ball
2. Return the ball to the urn
3. Add another ball of the same color to the urn

When $N$ is very large, the proportions of balls of different colors are distributed according to $\mathrm{Dirichlet}(\alpha )$.