Alternate explanation?
Let p = Probability of winning
Let q = Probability of not winning
As I recall from my statistics and probability classes, p + q =1.
If p = 1/3, then 1/3 + q = 1 -> q = 1 - 1/3 -> q = 2/3? In selecting your original door, your change of winning is p. By switching, your probability of winning is now q (or, rather, not p).
Just checking...