Yeah I remember when I heard about it years ago I actually wrote a simple program to show to myself that the second switching actually helps (yes, it does).
That’s not totally clear, due to the way the problem is often described: As a one-off event, with no clear rules for what the host will do. If there is no clear rule about what the host will do, you are still left in the dark. For example, the host may have decided beforehand (in secret) that he will open another door only if you picked the winning door. In that case, switching will lose you the prize with certainty.
But yes, if it is clearly understood that the host always opens another door after you picked one, you should switch.
But yes, if it is clearly understood that the host always opens another door after you picked one, you should switch.