TLDR: The statistics only work if the host has to reveal a goat and offer to let you switch.

In the show the question is based on, the host didn’t always open an incorrect door after a guess. He didn’t always allow them to switch. He also offered money instead of opening a door at all in some cases. He could use these tools to get the outcome he wanted most of the time.

tldr: The original problem statement does not say that the game host always has to open a losing door.

Really grea