Yes, it could work with more than three doors. In fact, the more doors there are, the better it is to switch your choice. Let's say we have five doors and there's only still one grand prize. If you pick a door at random, you have a 20% chance of picking the right door and an 80% chance of picking a wrong door. If the host then opens one of the other doors and shows you a goat, he is giving you some information that changes the probabilities of where the grand prize is. But how much do they change?

Well, think about it this way: when you first pick a door, there are two possible scenarios:

- Scenario A: You picked the right door (20% chance). In this case, it doesn't matter which door the host opens, because all of them have goats behind them. If you switch your choice, you will lose the grand prize and get a goat.
- Scenario B: You picked a wrong door (80% chance). In this case, the host has to open one of the other wrong doors, because he can't open your door or the right door. If you switch your choice, you will get the grand prize.

So switching your choice only helps you if you picked a wrong door in the first place, which happens 80% of the time. Therefore, switching your choice gives you an 80% chance of winning the grand prize. On the other hand, sticking with your original choice only helps you if you picked the right door in the first place, which happens 20% of the time. Therefore, sticking with your original choice gives you a 20% chance of winning the grand prize.

You can see that switching your choice improves your chances of winning by four times compared to sticking with your original choice. This is because when there are more doors, there is a higher probability that you picked a wrong door initially, and a lower probability that you picked the right door initially. When the host opens one of the other doors and shows you a goat, he is basically telling you that one of the remaining doors has a higher probability of having the grand prize behind it than your original choice.

I hope this explanation helps you understand why switching your choice in the Monty Hall problem with more than three doors is better than sticking with your original choice. If you have any questions or comments, please let me know.
spam fails_task lang_mismatch pii not_appropriate hate_speech sexual_content quality toxicity humor helpfulness creativity violence