三门问题

This is one version of the famous "Let's Make a Deal" or "Monty Hall" game show question. It is your turn to be on a weekly game show. There are three doors. You know that there is a prize behind one of them, and nothing behind the other two. The game show host tells you that you shall receive whatever is behind the door of your choice. However, before you choose, he tells you that he knows the actual location of

the prize, and he promises you that rather than immediately opening the door of your choice to reveal its contents, he will first open one of the other two doors to reveal that it is empty. He will then give you the option to change your mind and instead choose the remaining door that he did not open.

You may assume that whoever set up the doors and prizes placed the prize uniformly randomly behind a door (i.e., each door had an equal probability of being chosen as the prize location). You may assume that if you initially choose a door that has the prize, then the host is uniformly random in revealing one of the two remaining doors as empty. You may assume that the host must reveal an empty door.¹

You choose Door 3. He opens Door 2 and reveals that it is empty. You now know that the prize lies behind either Door 3 or Door 1. Should you switch your choice to Door 1?

I strongly recommend that you not look at the answer until you have done your best.