[theK]

Pretty sure the variables the author picked are not the most interesting ones.

Urn models are engineered to have a rich get richer bias which is best seen by varying the initial populations.

Instead of offering trial count and pick counts which are (invariates in the actual model) he could have picked initial ball count and initial white/red ratio.

[cmoog]

Ah, good idea! I'll add those in a bit and will remark that the answer/proof are specific for the special case where r_0 = b_0 = 1.