Please consider another view which is similar to what I posted 2 posts ago.
If you are the player. You already know how the host will conduct the game because you have seen it before.
You want to make a decision before you go on stage as to whether you are going to make the switch when offered, and stick to your decision.
Would you decide on switching or not switching as you strategy?
Does that change your perspective now?
Because you know that your chances of choosing the box with the key the first time is only 1/3, you will have a 2/3 chance of winning if you switch, and a 1/3 chance of winning if you do not switch. So does switching sounds like a better strategy to improve your chances of winning?
There is a subtle difference between making the switch from the initial choice, versus making a random choice the second time.
What I have been trying to show is that they are different, and those who believe that there is no difference whether or not the player switch base their conclusion on the notion that the second decision is a random choice between the 2 boxes versus a calculated decision on whether or not to make the switch.