Cumulative probabilities do not exist in reality. Like in numerology, you can add values all you want and you can asign numbers to each event and add them to succesive events with impecable correctness. But no matter what magical number or fraction you vest a box with, EACH event is unique.
The phase of the event that involves switching boxes is NOT a seperate independent event. It is part of the same event because, as I already explained, the selection choices in the second phase are directly shaped by the selection choices in the first phase. You can disentangle the two phases into separate events if you, as I suggested, blindfold yourself, spin around, and come back to the table not knowing which box you chose originally. Then you wouldn't know if you are switching or not, and you are starting afresh. But if you already know what you originally chose, then you know that it is more likely that the box you are holding is empty than that it contains the $$$. It's as simple as that.
I agree with Gerard. Although the first choice odds were 1/3, the opening of the empty box changed those odds. They are now 50/50. Switching does not improve those odds. In flipping a coin it doesn't matter whether you flip once or 100 times, the odds on the next flip is still 50/50. The coin doesn't remember past results.
This isn't a matter of inanimate objects remembering past results. Each flip of the coin is a discrete independent event. Each trial of this box-opening game is a discrete independent event. But as explained above, the first phase of the box-opening game is not independent of the second phase. Since the host cannot open the box containing the money, the mere act of selecting a box forces him to reveal one of two empty boxes which two-thirds of the time will correspond to an empty box that you are holding. One-third of the time you are holding the box containing the money and the host can freely choose either of the two other boxes. But two-thirds of the time, the host cannot freely pick a remaining box....he must pick one specific box out of the three. He has no choice. And that situation occurs twice as often as the instances when he can freely choose between the two boxes. So there is a connection between "events" (or rather phases of the same event) because the host's action is directly constrained by your prior action two-thirds of the time. When you switch boxes, you do so knowing that there was a greater chance than otherwise that host has opened the only other empty box. If you don't switch boxes, your odds are still one out of three (i.e. not 50/50) because your choice was made on the basis of three options and opening the boxes one at a time rather than all at once does not alter the odds of your choice.