  Drawing (a multiset of) coloured balls from an urn is one of the most basic
models in discrete probability theory. Three modes of drawing are commonly
distinguished: multinomial (draw-replace), hypergeometric (draw-delete), and
P\'olya (draw-add). These drawing operations are represented as maps from urns
to distributions over multisets of draws. The set of urns is a metric space via
the Wasserstein distance. The set of distributions over draws is also a metric
space, using Wasserstein-over-Wasserstein. It is shown that these three draw
operations are all isometries, that is, they exactly preserve the Wasserstein
distances. Further, drawing is studied in the limit, both for large urns and
for large draws. First it is shown that, as the urn size increases, the
Wasserstein distances go to zero between hypergeometric and multinomial draws,
and also between P\'olya and multinomial draws. Second, it is shown that, as
the draw size increases, the Wasserstein distance goes to zero (in probability)
between an urn and (normalised) multinomial draws from the urn. These results
are known, but here, they are formulated in a novel manner as limits of
Wasserstein distances. We call these two limit results the law of large urns
and the law of large draws.
