Why is Australia mostly desert?

[Ghostman]

In the case of the northern hemisphere deserts will be to the east of mountains due to Coriolis forces (forces due to the rotation of the spherical earth); in the southern hemisphere they will be on the west sides of mountains. In Australia, there is a lovely coastal range on the eastern edge of the continent with grassland grading to desert downwind of it.

Afraid you made a little mistake here. There is no difference between northern and southern hemispheres as far as the prevailing wind directions (and hence, direction of deserts relative to mountain ranges) goes.

It is in the equatorial region (roughly between the Tropics) that wind blows from east to west, potentially causing desertification to the west from mountains. When you get far enough from this region (either to the north or south) the winds will generally blow from west to east, and now the deserts will tend to be to the east from mountains. See the attached diagram for a visual explanation.