How often are two random integers between one and three, not the same?

The Monty Hall problem/answer = How often are two random integers between one and three (boundary inclusive), NOT the same. Can this be the clearest way to distill down the Monty Hall problem/answer down to its essence? I contend it can be, because of the fact that it only makes sense to NOT shift to the remaining door, if you actually happened to have guessed correctly initially, but that only happens 33% of the time, so you are always advantaged by switching. So in attempting to distill that down even further, success rate in shifting to the remaining door would be the same as the probability of two random integers between one and three NOT being the same, which would be 2/3. For me the above was the lightbulb in my understanding of the problem, and let to the further distillation summarized by the opening paragraph.